课程： 5.9 矢量方向 | The Way To Learn

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矢向方向
Alyssa loves physics, and it fascinates her that there are calculations for all of the physical interactions and motions in the world around her. On Friday, Alyssa went to the county fair with her boyfriend Kurt, who decided it would be romantic if he could win her a giant stuffed animal in the can knockdown game. Unfortunately for Kurt, he still had not knocked anything over approximately 45 mins and $35 later.
::Alyssa热爱物理,这令她着迷,因为全世界都在计算她周围的所有物理互动和运动。星期五,Alyssa和她的男朋友Kurt去县集,他决定如果他能赢得她一个巨型填充动物在罐子里的击倒游戏,那将是浪漫的。不幸的是,对于Kurt来说,他还没有敲过45分钟和35美元。

Alyssa decides to give it a try herself, and realizes that the center of the cans is a single position in space. If Alyssa knows her own throwing height, and the distance that the cans are stacked behind the throwing line, what else would she need to know to calculate the vectors involved with a straight-line path for the ball to take? How would she calculate them?
::Alyssa决定自己试一试,并意识到罐子的中心是空间中的一个位置。如果Alyssa知道她自己的投掷高度,以及罐子堆在抛出线后面的距离,她还需要知道什么才能计算出球直线路径所涉及的矢量?她要如何计算这些矢量呢?

Vector Direction
::矢向方向

Recall that the equation for a vector is given by
::回顾矢量的方方程式是由

$\begin{aligned} \vec{p} = ⟨ P_{x}, P_{y}, P_{z} ⟩ \end{aligned}$
::PPx,PY,PZZ

where P _x , P _y , and P _z are the x, y, and z coordinates of the vector obtained by projecting the vector onto the x, y, and z axes as shown below to the left.
::Px、Py和Pz是将矢量投射到 x、y 和 z 轴的矢量的 x、y 和 z 坐标,如下文所示,向左。

The image above to the right, shows the angles between the position vector, $\begin{aligned} \vec{P} \end{aligned}$ , and the three axes: α is the angle between $\begin{aligned} \vec{P} \end{aligned}$ and the x-axis, β is the angle between $\begin{aligned} \vec{P} \end{aligned}$ and the y-axis, and γ is the angle between $\begin{aligned} \vec{P} \end{aligned}$ and the z-axis.
::图像向右, 显示位置矢量、 P 和三个轴之间的角 : α 是 P 和 x 轴之间的角 , β 是 P 和 y 轴之间的角 , γ 是 P 和 z 轴之间的角。

The position vector, $\begin{aligned} \vec{P} \end{aligned}$ , and the unit vector, $\begin{aligned} \hat{x} \end{aligned}$ , define a plane shown in lavender below.
::位置矢量, P, 单位矢量, x, 定义一个平面, 在下面的熏衣草中显示。

The direction angle, α, is the angle between $\begin{aligned} \vec{P} \end{aligned}$ and $\begin{aligned} \hat{x} \end{aligned}$ in the plane defined by the two vectors. The other plane shown in the diagram is the X-Y plane, which was included in the diagram to help you visualize the orientation of the plane defined by the vectors.
::方向角α是两个矢量定义的平面中P和 x之间的角。图中显示的另一平面是 X-Y 平面,该平面包括在图中,以帮助您直观矢量定义的平面方向。

In our discussion of the dot product, we saw that the dot product of two vectors can be given by $\begin{aligned} \vec{A} \times \vec{B} = | \vec{A} | | \vec{B} | cos θ \end{aligned}$
::在讨论点产物时,我们看到两个矢量的点产物可以由ABABBCos提供。

Therefore, we can calculate the angle between $\begin{aligned} \vec{P} \end{aligned}$ and the unit vector $\begin{aligned} \hat{x} \end{aligned}$ .
::因此,我们可以计算P 和单位矢量 x 之间的角。

$\begin{aligned} α = {cos}^{- 1} \frac{\vec{P} \times \hat{x}}{| \vec{P} |} \end{aligned}$
::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Similarly, the direction angles β and γ can be calculated using the equations
::同样,可使用方程计算方向角度β和γ。

$\begin{aligned} β = {cos}^{- 1} \frac{\vec{P} \times \hat{y}}{| \vec{P} |} \end{aligned}$ and $\begin{aligned} γ = {cos}^{- 1} \frac{\vec{P} \times \hat{z}}{| \vec{P} |} \end{aligned}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In some applications, such as astronomy and applied optics, the direction cosines are used at least as frequently as the directional angles themselves.
::在诸如天文学和应用光学等一些应用中,对正向余弦的利用至少与方向角度本身一样频繁。

$\begin{aligned} cos α = \frac{\vec{P} \times \hat{x}}{| \vec{P} |}, cos β \frac{\vec{P} \times \hat{y}}{| \vec{P} |}, \end{aligned}$ and $\begin{aligned} cos γ = \frac{\vec{P} \times \hat{z}}{| \vec{P} |} \end{aligned}$
::,

Pythagorean Property of Direction Cosines
::方向脊椎动物财产

An interesting property of direction cosines can be seen if we write the equations for the direction cosines in terms of the components of the position vector, P _x , P _y , and P _z and using the definition of the vector magnitude $\begin{aligned} | \vec{A} | = \sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}} \end{aligned}$
::如果我们用位置矢量、 Px、 Py 和 Pz 的组件, 并使用矢量质量定义 AAAx2+Ay2+Az2, 并用此定义, 写出方向余弦值的方程, 就可以看到方向余弦值的有趣的属性。

For example,
::例如,

$\begin{aligned} cos α = \frac{P_{x}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::COs PxPx2+Py2+Pz2

The other two directional cosines can be rewritten similarly:
::另外两个方向余弦可作类似的改写:

$\begin{aligned} cos β = \frac{P_{y}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$ and $\begin{aligned} cos γ = \frac{P_{z}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::COs PyPx2+Py2+Pz2和COs PzPx2+Py2+Pz2

If we square both sides of all three equations and then sum them, we obtain
::如果我们在所有三个方方程式的两面都平开, 然后把它们加起来, 我们就能得到

$\begin{aligned} {cos}^{2} α + {cos}^{2} β + {cos}^{2} γ = \frac{P_{x}^{2}}{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}} + \frac{P_{y}^{2}}{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}} + \frac{P_{z}^{2}}{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}} \end{aligned}$
::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

which simplifies to
::简化到

$\begin{aligned} {cos}^{2} α + {cos}^{2} β + {cos}^{2} γ = \frac{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}} = 1 \end{aligned}$
::CO2+Pz2+Py2+Py2+Py2+Py2+Py2+Py2+Pz2=1

This is an important result because the definition of the unit vector stated that $\begin{aligned} | \hat{u} | = 1 \end{aligned}$
::这是一个重要的结果,因为单位矢量的定义表明,u1

which also means that
::也意味着

u _x ² + u _y ² + u _z ² = cos ² α + cos ² β + cos ² γ
::ux2 + uy2 + uz2 = cos2α + cos2β + cos2γ

and that the components of the unit vectors correspond to the direction cosines.
::单位矢量的组件对应方向余弦。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Alyssa and the can knockdown game.
::早些时候,你得到一个问题关于Alyssa 和罐头击倒游戏。

If Alyssa knows her own throwing height, and the distance that the cans are stacked behind the throwing line, (a) what else would she need to know to calculate the vectors involved with a straight-line path for the ball to take? (b) How would she calculate them?
::如果Alyssa知道她自己的抛掷高度, 以及罐子堆在抛掷线后面的距离, (a) 她还需要知道什么来计算与球直线路径相关的矢量? (b) 她如何计算这些矢量?

Alyssa needs to know the height of the cans above the ground.
::Alyssa需要知道地面上罐头的高度

If Alyssa assumes that the center of the cans is directly in front of her, and treats her own hand as the origin, she can calculate the vector directions as above, using 0 for x , the difference between her throwing height and that of the cans as y , and the distance that the cans are stacked beyond the throwing line as z .
::如果Alyssa认为罐头的中心直接在她面前, 并且将她自己的手作为源头, 她可以计算上面的矢量方向, 使用 0 的 x, 她投掷的高度和罐头之间的差别是 Y, 以及罐头堆在抛出线外的距离是 z 。

Example 2
::例2

Because airplanes move in three dimensions, air-craft ground crews can use direction cosines to identify their location at any moment. Rather than an arbitrary set of orthogonal x, y, and z axes, the position of an airplane is measured relative to the east, north, and zenith directions. (Zenith means top or upward.) At a particular moment, a small plane is 297 km east, 135 km north, and 7.5 km above its home airport. What are the directional cosines and directional angles for the position vector of the plane at that moment?
::由于飞机在三个维度上移动, 空勤地面机组人员可以使用方向脉冲来识别其位置。飞机的位置不是任意的直角 x、 y 和 z 轴,而是相对于东、北和天顶方向测量的。 (天顶是指上或上。 ) 在某个特定时刻, 一架小飞机位于东297公里、北135公里、上方机场上方7.5公里。飞机的位置矢量的方向脉冲和方向角度是什么?

If we use the standard orientation of maps in the northern hemisphere, the x-direction corresponds to east, the y-direction to north, and the z-direction to the zenith. Therefore, the position vector of the plane can be written as $\begin{aligned} \vec{P} = ⟨ 297, 135, 7.5 ⟩ km \end{aligned}$ .
::如果我们使用北半球地图的标准方向,x方向与东对齐,Y方向与北对齐,Z方向与天顶对齐。因此,飞机的方位矢量可以写为P 297,135.7.5km。

The direction cosines associated with this vector are given by
::与此向量相关联的余弦由

$\begin{aligned} cos α = \frac{\vec{P} \times \hat{x}}{| \vec{P} |} = \frac{P_{x}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::QPxPPPxPx2+Py2+Pz2

$\begin{aligned} cos α = \frac{P_{x}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} = \frac{297}{\sqrt{297^{2} + 135^{2} + {7.5}^{2}}} = 0.582 \end{aligned}$
::PxPx2+Py2+Pz2=2972972+1352+7.52=0.582

$\begin{aligned} cos β = \frac{\vec{P} \times \hat{y}}{| \vec{P} |} = \frac{P_{y}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::COs PPPPPPyPx2+Py2+Pz2+Pz2+Py2+Py2+Py2+Py2+Py2+PyPyPyPx2+

$\begin{aligned} cos β = \frac{P_{y}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} = \frac{135}{\sqrt{297^{2} + 135^{2} + {7.5}^{2}}} = 0.811 \end{aligned}$
::PyPx2+Py2+Pz2=1352972+13552+7.52=0.811

$\begin{aligned} cos γ = \frac{\vec{P} \times \hat{z}}{| \vec{P} |} = \frac{P_{z}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::PzPx2+Py2+Pz2

$\begin{aligned} cos γ = \frac{P_{z}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} = \frac{7500}{\sqrt{297^{2} + 135^{2} + {7.5}^{2}}} = 0.045 \end{aligned}$
::PzPx2+Py2+Pz2=75002972+1352+7.52=0.045

The associated direction angles are
::相关方向角度为

α = cos ^-1 0.582 = 54.4 ^o
::α = COS-1 0.582 = 54.4o

β = cos ^-1 0.165 = 35.8 ^o
::β = COS-1 0165 = 35.8o

γ = cos ^-1 0.045 = 87.4 ^o
::γ = COS-1 0.045 = 87.4o

Example 3
::例3

In an upcoming episode of a crime drama, a swarm of special-effect insects bothers one of the investigators after the discovery of a murder victim in a drainage ditch. The animator uses position vectors to track the positions of the virtual pests with respect to an origin at the investigator’s head. One such insect is located at a point 33 cm in front, 52 cm to the left, and 18 cm below the tip of the investigator’s nose. What are the directional cosines for this insect?
::在即将到来的一场犯罪剧中,在发现下水道中一名被害者后,一群特效昆虫骚扰了一名调查人员。动画师使用定位矢量跟踪虚拟害虫相对于调查员头部来源的位置。其中一个昆虫位于前方33厘米、左侧52厘米和下方18厘米处。这个昆虫的指向系是什么?

As we look at our investigator, the +x direction is to her left, the +y direction is upward from her nose, and the +z direction is in front of her. The position vector of the midge can be written as $\begin{aligned} \vec{P} = ⟨ 33, 52, - 18 ⟩ cm \end{aligned}$ .
::当我们看我们的调查员时, +x方向在她的左边, +y方向在她的鼻子上方, +z方向在她的前面。蚊子的位置矢量可以写为 P33, 52, - 18 cmm 。

The direction cosines associated with this vector are given by
::与此向量相关联的余弦由

$\begin{aligned} cos α = \frac{\vec{P} \times \hat{x}}{| \vec{P} |} = \frac{P_{x}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::QPxPPPxPx2+Py2+Pz2

$\begin{aligned} cos α = \frac{P_{x}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} = \frac{33}{\sqrt{33^{2} + 52^{2} + (- 18)^{2}}} = 0.514 \end{aligned}$
::COs PxPx2+Py2+Pz2=33332+522+(-18)2=0.514

$\begin{aligned} cos β = \frac{\vec{P} \times \hat{y}}{| \vec{P} |} = \frac{P_{y}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::COs PPPPPPyPx2+Py2+Pz2+Pz2+Py2+Py2+Py2+Py2+Py2+PyPyPyPx2+

$\begin{aligned} cos β = \frac{P_{y}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} = \frac{52}{\sqrt{33^{2} + 52^{2} + (- 18)^{2}}} = 0.810 \end{aligned}$
::PyPx2+Py2+Pz2=52332+522+(-18)2=0.810

$\begin{aligned} cos γ = \frac{\vec{P} \times \hat{z}}{| \vec{P} |} = \frac{P_{z}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::PzPx2+Py2+Pz2

$\begin{aligned} cos γ = \frac{P_{z}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} = \frac{- 18}{\sqrt{33^{2} + 52^{2} + (- 18)^{2}}} = - 0.281 \end{aligned}$
::PzPx2+Py2+Pz218332+522+(-188)20.281

Example 4
::例4

A local astronomer used direction cosines when programming the projector in a new planetarium dome. The projector itself sits at the center of the dome, 2.5 m above the floor. He wishes to project Mintaka, one of the stars in the belt of Orion, at a position 12 m south and 2.3 m east of the projector and 8.7 m above the floor. What is the equation of the directional unit vector that the astronomer must enter in to the projection computer?
::一个当地天文学家在将投影器编程成一个新的天文馆圆顶时使用了方向焦距。投影器本身位于圆顶的中心,在地面上2.5米处。他想将Orion带的恒星之一Mintaka投影到投影机以南12米、投影机以东2.3米和上方8.7米处。天文学家必须进入投影计算机的方向单位矢量的方程是多少?

We can use the same coordinate system that we used in example A above: $\begin{aligned} \hat{x} \end{aligned}$ = east, $\begin{aligned} \hat{y} \end{aligned}$ = north, and $\begin{aligned} \hat{z} \end{aligned}$ = upward. The projector itself is the origin. In such a coordinate system, the position vector for Mintaka becomes $\begin{aligned} \vec{P} = ⟨ 2.3, - 12, (8.7 - 2.5) ⟩ = ⟨ 2.3, - 12, 6.2 ⟩ \end{aligned}$ . Notice that we did not use Mintaka’s position above the floor as the z-coordinate. Rather, since the projector is 2.5 m above the floor, we needed to use the difference between the ceiling height and the projector height. Use the component form of the directional cosine equation to calculate the three components of the unit vector.
::我们可以使用我们在上文例A中使用的相同的坐标系统: x = 东部, y = 北部, z = 向上。投影仪本身是源头。在这样的坐标系统中, Mintaka 的定位矢量成为 P2.3, -12, (8. 7- 2.5) 2.3, - 12, 6.2 。请注意, 我们没有使用 Mintaka 的地面位置作为Z坐标。相反, 由于投影仪在地面上方2.5米, 我们需要使用天花板高度和投影仪高度之间的差别。使用定向连弦方程式的组件来计算单元矢量的三个组成部分。

$\begin{aligned} cos α = \frac{P_{x}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} = \frac{2.3}{\sqrt{{2.3}^{2} + (- 12)^{2} + {6.2}^{2}}} = 0.168 \end{aligned}$
::PxPx2+Py2+Pz2=2.3.32+(-12)2+6.22=0.168

$\begin{aligned} cos β = \frac{P_{x}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} = \frac{- 12}{\sqrt{{2.3}^{2} + (- 12)^{2} + {6.2}^{2}}} = - 0.876 \end{aligned}$
::PxPx2+Py2+Pz2122.32+(-12)2+6.220.876

$\begin{aligned} cos γ = \frac{P_{x}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} = \frac{6.2}{\sqrt{{2.3}^{2} + (- 12)^{2} + {6.2}^{2}}} = 0.453 \end{aligned}$
::PxPx2+Py2+Pz2=6.22.32+(-12)2+6.22=0.453

Therefore, the position unit vector is given by $\begin{aligned} \hat{u} = ⟨ 0.168, - 0.876, 0.453 ⟩ \end{aligned}$ .
::因此,位置单位矢量由 u0.168 -0.876,0.453 提供。

Example 5
::例5

Determine the components of the position vector $\begin{aligned} \vec{P} = ⟨ 2.4, 5.3, 1.8 ⟩ \end{aligned}$ then determine the directional angles between this vector and the x-axis.
::确定位置矢量的组件 P2.4、5.3、1.8,然后确定此矢量与 x 轴之间的方向角度。

$\begin{aligned} cos α = \frac{\vec{P} \times \hat{x}}{| \vec{P} |} = \frac{P_{x}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::QPxPPPxPx2+Py2+Pz2

$\begin{aligned} cos α = \frac{\vec{P} \times \hat{x}}{| \vec{P} |} = \frac{2.4}{\sqrt{(2.4)^{2} + (5.3)^{2} + (1.8)^{2}}} = \frac{2.4}{\sqrt{5.76 + 28.09 + 3.24}} = \frac{2.4}{\sqrt{37.09}} = 0.394 \end{aligned}$
::+(5.3)2+(1.8)2=2.45.6.76+28.09+3.24=2.437.09=0.394

$\begin{aligned} α = {cos}^{- 1} 0.394 = 66.8 \end{aligned}$
::cos- 10.394=66.8

Example 6
::例6

Determine the direction cosines for the vector $\begin{aligned} \vec{N} = ⟨ 8, 3, - 5 ⟩ \end{aligned}$ .
::确定矢量N8、3、-5的正弦值方向。

$\begin{aligned} cos α = \frac{\vec{N} \times \hat{x}}{| \vec{N} |} = \frac{N_{x}}{\sqrt{N_{x}^{2} + N_{y}^{2} + N_{z}^{2}}} \end{aligned}$
::{NxNNNxNx2+Ny2+Nz2 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-58b"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Calpha%20%3D%20%5Cfrac%7B%5Coverrightarrow%7BN%7D%20%5Ctimes%20%5Chat%7Bx%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7BN%7D%20%7C%7D%20%3D%20%5Cfrac%7BN_x%7D%20%7B%5Csqrt%7BN%5E2_x%20%2B%20N%5E2_y%20%2B%20N%5E2_z%7D%7D%20%3D%20%5Cfrac%7B8%7D%20%7B%5Csqrt%7B(8)%5E2%20%2B%20(3)%5E2%20%2B%20(-5)%5E2%7D%7D%20%3D%20%5Cfrac%7B8%7D%20%7B%5Csqrt%7B64%20%2B%209%20%2B%2025%7D%7D%20%3D%200.7213"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>α</mi><mo>=</mo><mfrac><mrow><mover><mi>N</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>x</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>N</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>N</mi><mi>x</mi></msub><msqrt><msubsup><mi>N</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>N</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>N</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac><mo>=</mo><mfrac><mn>8</mn><msqrt><mo stretchy="false">(</mo><mn>8</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mn>3</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mfrac><mn>8</mn><msqrt><mn>64</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>25</mn></msqrt></mfrac><mo>=</mo><mn>0.7213</mn></mtd></mtr></mtable></math>' id="MathJax-Element-53-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="9.205ex" role="img" style="vertical-align: -4.028ex;" viewbox="0 -2229 39131.9 3963.1" width="90.887ex" xmlns:xlink="http://www.w3.org/1999/xlink"> <g fill="currentColor" role="group" stroke="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"> <g role="group" transform="translate(167,0)"> <g role="group" transform="translate(-16,0)"> <g role="group" transform="translate(0,183)"> <use xlink:href="#MJMAIN-63"> </use> <use x="444" xlink:href="#MJMAIN-6F" y="0"> </use> <use x="945" xlink:href="#MJMAIN-73" y="0"> </use> <use x="1589" 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y="0"> </use> <use x="779" xlink:href="#MJMAIN-37" y="0"> </use> <use x="1279" xlink:href="#MJMAIN-32" y="0"> </use> <use x="1780" xlink:href="#MJMAIN-31" y="0"> </use> <use x="2280" xlink:href="#MJMAIN-33" y="0"> </use> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> α </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> N </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> N </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> N </mi> <mi> x </mi> </msub> <msqrt> <msubsup> <mi> N </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> N </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> N </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mn> 8 </mn> <msqrt> <mo stretchy="false"> ( </mo> <mn> 8 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 3 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 5 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mn> 8 </mn> <msqrt> <mn> 64 </mn> <mo> + </mo> <mn> 9 </mn> <mo> + </mo> <mn> 25 </mn> </msqrt> </mfrac> <mo> = </mo> <mn> 0.7213 </mn> </mtd> </mtr> </mtable> </math> </span> </span> </span> <br/> <span style="color: green; "> ::=8(8)2+(3)2+(-5)2=864+9+25=0.7213 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-pdw"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Cbeta%20%3D%20%5Cfrac%7B%5Coverrightarrow%7BN%7D%20%5Ctimes%20%5Chat%7By%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7BN%7D%20%7C%7D%20%3D%20%5Cfrac%7BN_y%7D%20%7B%5Csqrt%7BN%5E2_x%20%2B%20N%5E2_y%20%2B%20N%5E2_z%7D%7D"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>β</mi><mo>=</mo><mfrac><mrow><mover><mi>N</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>N</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>N</mi><mi>y</mi></msub><msqrt><msubsup><mi>N</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>N</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>N</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac></mtd></mtr></mtable></math>' id="MathJax-Element-54-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="9.205ex" role="img" style="vertical-align: -4.028ex;" viewbox="0 -2229 16342.9 3963.1" width="37.958ex" 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xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> β </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> N </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> y </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> N </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> N </mi> <mi> y </mi> </msub> <msqrt> <msubsup> <mi> N </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> N </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> N </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> </mtd> </mtr> </mtable> </math> </span> </span> </span> <br/> <span style="color: green; "> ::{NYNNNyNx2+Ny2+Nz2 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-ptv"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Cbeta%20%3D%20%5Cfrac%7B%5Coverrightarrow%7BN%7D%20%5Ctimes%20%5Chat%7By%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7BN%7D%20%7C%7D%20%3D%20%5Cfrac%7BN_y%7D%20%7B%5Csqrt%7BN%5E2_x%20%2B%20N%5E2_y%20%2B%20N%5E2_z%7D%7D%20%3D%20%5Cfrac%7B3%7D%20%7B%5Csqrt%7B(8)%5E2%20%2B%20(3)%5E2%20%2B%20(-5)%5E2%7D%7D%20%3D%20%5Cfrac%7B3%7D%20%7B%5Csqrt%7B64%20%2B%209%20%2B%2025%7D%7D%20%3D%200.3030"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>β</mi><mo>=</mo><mfrac><mrow><mover><mi>N</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>N</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>N</mi><mi>y</mi></msub><msqrt><msubsup><mi>N</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>N</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>N</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac><mo>=</mo><mfrac><mn>3</mn><msqrt><mo stretchy="false">(</mo><mn>8</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mn>3</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mfrac><mn>3</mn><msqrt><mn>64</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>25</mn></msqrt></mfrac><mo>=</mo><mn>0.3030</mn></mtd></mtr></mtable></math>' id="MathJax-Element-55-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="9.205ex" role="img" style="vertical-align: -4.028ex;" viewbox="0 -2229 39053.1 3963.1" width="90.704ex" xmlns:xlink="http://www.w3.org/1999/xlink"> <g fill="currentColor" role="group" stroke="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"> <g role="group" transform="translate(167,0)"> <g role="group" transform="translate(-16,0)"> <g 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y="0"> </use> </g> </g> </g> </g> </g> <use x="34896" xlink:href="#MJMAIN-3D" y="0"> </use> <g role="group" transform="translate(35953,0)"> <use xlink:href="#MJMAIN-30"> </use> <use x="500" xlink:href="#MJMAIN-2E" y="0"> </use> <use x="779" xlink:href="#MJMAIN-33" y="0"> </use> <use x="1279" xlink:href="#MJMAIN-30" y="0"> </use> <use x="1780" xlink:href="#MJMAIN-33" y="0"> </use> <use x="2280" xlink:href="#MJMAIN-30" y="0"> </use> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> β </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> N </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> y </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> N </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> N </mi> <mi> y </mi> </msub> <msqrt> <msubsup> <mi> N </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> N </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> N </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mn> 3 </mn> <msqrt> <mo stretchy="false"> ( </mo> <mn> 8 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 3 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 5 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mn> 3 </mn> <msqrt> <mn> 64 </mn> <mo> + </mo> <mn> 9 </mn> <mo> + </mo> <mn> 25 </mn> </msqrt> </mfrac> <mo> = </mo> <mn> 0.3030 </mn> </mtd> </mtr> </mtable> </math> </span> </span> </span> <br/> <span style="color: green; "> ::=3(8)2+(3)2+(-5)2=364+9+25=0.3030 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-d6x"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Cgamma%20%3D%20%5Cfrac%7B%5Coverrightarrow%7BN%7D%20%5Ctimes%20%5Chat%7Bz%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7BN%7D%20%7C%7D%20%3D%20%5Cfrac%7BN_z%7D%20%7B%5Csqrt%7BN%5E2_x%20%2B%20N%5E2_y%20%2B%20N%5E2_z%7D%7D"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>γ</mi><mo>=</mo><mfrac><mrow><mover><mi>N</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>z</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>N</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>N</mi><mi>z</mi></msub><msqrt><msubsup><mi>N</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>N</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>N</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac></mtd></mtr></mtable></math>' id="MathJax-Element-56-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg 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</g> </g> </g> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> γ </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> N </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> z </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> N </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> N </mi> <mi> z </mi> </msub> <msqrt> <msubsup> <mi> N </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> N </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> N </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> </mtd> </mtr> </mtable> </math> </span> </span> </span> <br/> <span style="color: green; "> ::{NZNNNzNx2+Ny2+Nz2 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-w6p"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Cgamma%20%3D%20%5Cfrac%7B%5Coverrightarrow%7BN%7D%20%5Ctimes%20%5Chat%7Bz%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7BN%7D%20%7C%7D%20%3D%20%5Cfrac%7BN_z%7D%20%7B%5Csqrt%7BN%5E2_x%20%2B%20N%5E2_y%20%2B%20N%5E2_z%7D%7D%20%3D%20%5Cfrac%7B-5%7D%20%7B%5Csqrt%7B(8)%5E2%20%2B%20(3)%5E2%20%2B%20(-5)%5E2%7D%7D%20%3D%20%5Cfrac%7B-5%7D%20%7B%5Csqrt%7B64%20%2B%209%20%2B%2025%7D%7D%20%3D%20-0.5051"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>γ</mi><mo>=</mo><mfrac><mrow><mover><mi>N</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>z</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>N</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>N</mi><mi>z</mi></msub><msqrt><msubsup><mi>N</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>N</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>N</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>5</mn></mrow><msqrt><mo stretchy="false">(</mo><mn>8</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mn>3</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>5</mn></mrow><msqrt><mn>64</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>25</mn></msqrt></mfrac><mo>=</mo><mo>−</mo><mn>0.5051</mn></mtd></mtr></mtable></math>' id="MathJax-Element-57-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="9.205ex" role="img" style="vertical-align: -4.028ex;" viewbox="0 -2229 39799 3963.1" width="92.437ex" xmlns:xlink="http://www.w3.org/1999/xlink"> <g fill="currentColor" role="group" stroke="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"> <g role="group" transform="translate(167,0)"> <g role="group" transform="translate(-16,0)"> <g role="group" transform="translate(0,183)"> <use xlink:href="#MJMAIN-63"> </use> <use x="444" xlink:href="#MJMAIN-6F" y="0"> </use> <use x="945" xlink:href="#MJMAIN-73" y="0"> </use> <use x="1589" xlink:href="#MJMATHI-3B3" y="0"> </use> <use x="2410" xlink:href="#MJMAIN-3D" y="0"> </use> <g role="group" transform="translate(3189,0)"> <g role="group" transform="translate(397,0)"> <rect height="60" stroke="none" width="3040" x="0" y="220"> </rect> <g role="group" transform="translate(60,676)"> <use x="56" xlink:href="#MJMATHI-4E" y="0"> </use> <use x="138" xlink:href="#MJMAIN-2192" y="770"> </use> <use x="1361" 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</g> <g role="group" transform="translate(60,-845)"> <use x="0" xlink:href="#MJMAIN-221A" y="20"> </use> <rect height="60" stroke="none" width="4948" x="833" y="761"> </rect> <g role="group" transform="translate(833,0)"> <use xlink:href="#MJMAIN-36"> </use> <use x="500" xlink:href="#MJMAIN-34" y="0"> </use> <use x="1223" xlink:href="#MJMAIN-2B" y="0"> </use> <use x="2223" xlink:href="#MJMAIN-39" y="0"> </use> <use x="2946" xlink:href="#MJMAIN-2B" y="0"> </use> <g role="group" transform="translate(3947,0)"> <use xlink:href="#MJMAIN-32"> </use> <use x="500" xlink:href="#MJMAIN-35" y="0"> </use> </g> </g> </g> </g> </g> <use x="34864" xlink:href="#MJMAIN-3D" y="0"> </use> <use x="35920" xlink:href="#MJMAIN-2212" y="0"> </use> <g role="group" transform="translate(36699,0)"> <use xlink:href="#MJMAIN-30"> </use> <use x="500" xlink:href="#MJMAIN-2E" y="0"> </use> <use x="779" xlink:href="#MJMAIN-35" y="0"> </use> <use x="1279" xlink:href="#MJMAIN-30" y="0"> </use> <use x="1780" xlink:href="#MJMAIN-35" y="0"> </use> <use x="2280" xlink:href="#MJMAIN-31" y="0"> </use> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> γ </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> N </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> z </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> N </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> N </mi> <mi> z </mi> </msub> <msqrt> <msubsup> <mi> N </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> N </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> N </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mrow> <mo> − </mo> <mn> 5 </mn> </mrow> <msqrt> <mo stretchy="false"> ( </mo> <mn> 8 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 3 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 5 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mrow> <mo> − </mo> <mn> 5 </mn> </mrow> <msqrt> <mn> 64 </mn> <mo> + </mo> <mn> 9 </mn> <mo> + </mo> <mn> 25 </mn> </msqrt> </mfrac> <mo> = </mo> <mo> − </mo> <mn> 0.5051 </mn> </mtd> </mtr> </mtable> </math> </span> </span> </span> <br/> <span style="color: green; "> ::===================================================================================================================================================== </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <h4 id="x-ck12-ZjVjMmRkYmZiOWFkY2FiZGUwZmQ1NGIwODI2NmQ0ZTc.-cgl"> Example 7 <br/> <span style="color: green; "> ::例7 </span> </h4> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-NGNlNDhkODkxNTljZmNiY2M0ZDIzZGZkOGJmNjA4ODI.-6qp"> Determine the position vector for a small plane at the moment it is 2.5 km east, 8.8 km south, and 4.1 km above its home airport. Use a coordinate system where the x-direction corresponds to east, the y-direction to north, and the z-direction to the zenith. Then determine the directional cosines used by the air-traffic control personnel to identify the plane’s location. <br/> <span style="color: green; "> ::确定一架小飞机在东经2.5公里、南经8.8公里、离本国机场4.1公里处的位置矢量。使用一个坐标系统,使X向对准东,Y向向向北,Z向向天顶。然后确定空中交通管制人员用来确定飞机位置的方向余弦。 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-YTNiMDFjNjgxY2Y0OTEzODg5N2UzMDJmOTJmOWMzZjc.-ts5"> In this coordinate system, with an origin at the home airport, the position vector is given by <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Coverrightarrow%7Br%7D%20%3D%20%5Cleft%20%5Clangle%202.5%2C%20-8.8%2C%204.1%5Cright%20%5Crangle"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mover><mi>r</mi><mo>→</mo></mover><mo>=</mo><mrow><mo>⟨</mo><mrow><mn>2.5</mn><mo>,</mo><mo>−</mo><mn>8.8</mn><mo>,</mo><mn>4.1</mn></mrow><mo>⟩</mo></mrow></mtd></mtr></mtable></math>' id="MathJax-Element-58-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="3.458ex" role="img" style="vertical-align: -1.154ex;" viewbox="0 -991.8 8995.3 1488.7" width="20.892ex" xmlns:xlink="http://www.w3.org/1999/xlink"> <g fill="currentColor" role="group" stroke="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"> <g role="group" transform="translate(167,0)"> <g role="group" transform="translate(-16,0)"> <g role="group" transform="translate(0,-146)"> <use x="274" xlink:href="#MJMATHI-72" y="0"> </use> <use x="55" xlink:href="#MJMAIN-2192" y="530"> </use> <use x="1333" xlink:href="#MJMAIN-3D" y="0"> </use> <g role="group" transform="translate(2390,0)"> <use x="0" xlink:href="#MJMAIN-27E8" y="0"> </use> <g role="group" transform="translate(389,0)"> <use xlink:href="#MJMAIN-32"> </use> <use x="500" xlink:href="#MJMAIN-2E" y="0"> </use> <use x="779" xlink:href="#MJMAIN-35" y="0"> </use> <use x="1279" xlink:href="#MJMAIN-2C" y="0"> </use> <use x="1724" xlink:href="#MJMAIN-2212" y="0"> </use> <g role="group" transform="translate(2503,0)"> <use xlink:href="#MJMAIN-38"> </use> <use x="500" xlink:href="#MJMAIN-2E" y="0"> </use> <use x="779" xlink:href="#MJMAIN-38" y="0"> </use> </g> <use x="3782" xlink:href="#MJMAIN-2C" y="0"> </use> <g role="group" transform="translate(4227,0)"> <use xlink:href="#MJMAIN-34"> </use> <use x="500" xlink:href="#MJMAIN-2E" y="0"> </use> <use x="779" xlink:href="#MJMAIN-31" y="0"> </use> </g> </g> <use x="5896" xlink:href="#MJMAIN-27E9" y="0"> </use> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mover> <mi> r </mi> <mo> → </mo> </mover> <mo> = </mo> <mrow> <mo> ⟨ </mo> <mrow> <mn> 2.5 </mn> <mo> , </mo> <mo> − </mo> <mn> 8.8 </mn> <mo> , </mo> <mn> 4.1 </mn> </mrow> <mo> ⟩ </mo> </mrow> </mtd> </mtr> </mtable> </math> </span> </span> </span> with units of kilometers. <br/> <span style="color: green; "> ::在这一协调系统中,位置矢量由单位为公里的r2.5,-8.8,4.1提供,原产于本国机场,位置矢量由单位为公里。 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-zcg"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Calpha%20%3D%20%5Cfrac%7B%5Coverrightarrow%7Br%7D%20%5Ctimes%20%5Chat%7Bx%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7Br%7D%20%7C%7D%20%3D%20%5Cfrac%7Br_x%7D%20%7B%5Csqrt%7Br%5E2_x%20%2B%20r%5E2_y%20%2B%20r%5E2_z%7D%7D"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>α</mi><mo>=</mo><mfrac><mrow><mover><mi>r</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>x</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>r</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>r</mi><mi>x</mi></msub><msqrt><msubsup><mi>r</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>r</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>r</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac></mtd></mtr></mtable></math>' id="MathJax-Element-59-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="8.515ex" role="img" style="vertical-align: -3.683ex;" viewbox="0 -2080.5 15002 3666.2" width="34.844ex" xmlns:xlink="http://www.w3.org/1999/xlink"> <g fill="currentColor" role="group" stroke="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"> <g role="group" transform="translate(167,0)"> <g role="group" transform="translate(-16,0)"> <g role="group" transform="translate(0,303)"> <use xlink:href="#MJMAIN-63"> </use> <use x="444" xlink:href="#MJMAIN-6F" 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</use> </g> </g> </g> <use x="7053" xlink:href="#MJMAIN-3D" y="0"> </use> <g role="group" transform="translate(7831,0)"> <g role="group" transform="translate(397,0)"> <rect height="60" stroke="none" width="6333" x="0" y="220"> </rect> <g role="group" transform="translate(2688,676)"> <use x="0" xlink:href="#MJMATHI-72" y="0"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMATHI-78" y="-213"> </use> </g> <g role="group" transform="translate(60,-1209)"> <use x="0" xlink:href="#MJSZ2-221A" y="33"> </use> <rect height="60" stroke="none" width="5213" x="1000" y="1124"> </rect> <g role="group" transform="translate(1000,0)"> <use x="0" xlink:href="#MJMATHI-72" y="0"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMAIN-32" y="488"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMATHI-78" y="-211"> </use> <use x="1178" xlink:href="#MJMAIN-2B" y="0"> </use> <g role="group" transform="translate(2179,0)"> <use x="0" xlink:href="#MJMATHI-72" y="0"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMAIN-32" y="488"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMATHI-79" y="-212"> </use> </g> <use x="3306" xlink:href="#MJMAIN-2B" y="0"> </use> <g role="group" transform="translate(4307,0)"> <use x="0" xlink:href="#MJMATHI-72" y="0"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMAIN-32" y="488"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMATHI-7A" y="-211"> </use> </g> </g> </g> </g> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> α </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> r </mi> <mi> x </mi> </msub> <msqrt> <msubsup> <mi> r </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> </mtd> </mtr> </mtable> </math> </span> </span> </span> <br/> <span style="color: green; "> ::+ry2+rz2 +rz2 + ry2+rz2+rxx2+ry2+ry2+rz2+rz2+rz2+rz2+rxrx2+rxx2+ry2+ry2+rz2+rz2+rxx2+rx2+rz2+rz </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-0kl"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Calpha%20%3D%20%5Cfrac%7B%5Coverrightarrow%7Br%7D%20%5Ctimes%20%5Chat%7Bx%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7Br%7D%20%7C%7D%20%3D%20%5Cfrac%7Br_x%7D%20%7B%5Csqrt%7Br%5E2_x%20%2B%20r%5E2_y%20%2B%20r%5E2_z%7D%7D%20%3D%20%5Cfrac%7B2.5%7D%20%7B%5Csqrt%7B(2.5)%5E2%20%2B%20(-8.8)%5E2%20%2B%20(4.1)%5E2%7D%7D%20%3D%20%5Cfrac%7B2.5%7D%20%7B%5Csqrt%7B100.5%7D%7D%20%3D%200.249"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>α</mi><mo>=</mo><mfrac><mrow><mover><mi>r</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>x</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>r</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>r</mi><mi>x</mi></msub><msqrt><msubsup><mi>r</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>r</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>r</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac><mo>=</mo><mfrac><mn>2.5</mn><msqrt><mo stretchy="false">(</mo><mn>2.5</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>8.8</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mn>4.1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mfrac><mn>2.5</mn><msqrt><mn>100.5</mn></msqrt></mfrac><mo>=</mo><mn>0.249</mn></mtd></mtr></mtable></math>' id="MathJax-Element-60-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="8.515ex" role="img" style="vertical-align: -3.683ex;" viewbox="0 -2080.5 36880.8 3666.2" width="85.659ex" xmlns:xlink="http://www.w3.org/1999/xlink"> <g fill="currentColor" role="group" stroke="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"> <g role="group" transform="translate(167,0)"> <g role="group" transform="translate(-16,0)"> <g role="group" transform="translate(0,303)"> <use xlink:href="#MJMAIN-63"> </use> <use x="444" xlink:href="#MJMAIN-6F" y="0"> </use> <use x="945" xlink:href="#MJMAIN-73" y="0"> </use> <use x="1589" xlink:href="#MJMATHI-3B1" y="0"> </use> <use x="2507" xlink:href="#MJMAIN-3D" y="0"> </use> 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displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> α </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> r </mi> <mi> x </mi> </msub> <msqrt> <msubsup> <mi> r </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mn> 2.5 </mn> <msqrt> <mo stretchy="false"> ( </mo> <mn> 2.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 8.8 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 4.1 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mn> 2.5 </mn> <msqrt> <mn> 100.5 </mn> </msqrt> </mfrac> <mo> = </mo> <mn> 0.249 </mn> </mtd> </mtr> </mtable> </math> </span> </span> </span> <br/> <span style="color: green; "> ::=2.5(2.5)2+(-8.8)2+(4.1)2=2.5100.5=0.249 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-puc"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Cbeta%20%3D%20%5Cfrac%7B%5Coverrightarrow%7Br%7D%20%5Ctimes%20%5Chat%7By%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7Br%7D%20%7C%7D%20%3D%20%5Cfrac%7Br_y%7D%20%7B%5Csqrt%7Br%5E2_x%20%2B%20r%5E2_y%20%2B%20r%5E2_z%7D%7D"> <span class="MathJax_Preview" style="color: 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x="638" xlink:href="#MJMAIN-32" y="488"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMATHI-79" y="-212"> </use> </g> <use x="3306" xlink:href="#MJMAIN-2B" y="0"> </use> <g role="group" transform="translate(4307,0)"> <use x="0" xlink:href="#MJMATHI-72" y="0"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMAIN-32" y="488"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMATHI-7A" y="-211"> </use> </g> </g> </g> </g> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> β </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> y </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> r </mi> <mi> y </mi> </msub> <msqrt> <msubsup> <mi> r </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> </mtd> </mtr> </mtable> </math> </span> </span> </span> <br/> <span style="color: green; "> ::+ry2+rz2+rz2+ry2+rz2+ry2+rrrrx2+ry2+rz2+rz2+rz2+rz2+ry2+ry2+rrrx2+ry2+ry2+rrrx2+ry2+rrrx2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rx2+rx2+ry2+ry2+ry2+rx2+ry2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+r2+rz2+r2+rz2+rz2+r2+z2+r2+rz2+r2+r2+r2+r2+r2+r2+r2+r2+r2+r2+z2+z </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-9fw"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Cbeta%20%3D%20%5Cfrac%7B%5Coverrightarrow%7Br%7D%20%5Ctimes%20%5Chat%7By%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7Br%7D%20%7C%7D%20%3D%20%5Cfrac%7Br_y%7D%20%7B%5Csqrt%7Br%5E2_x%20%2B%20r%5E2_y%20%2B%20r%5E2_z%7D%7D%20%3D%20%5Cfrac%7B-8.8%7D%20%7B%5Csqrt%7B(2.5)%5E2%20%2B%20(-8.8)%5E2%20%2B%20(4.1)%5E2%7D%7D%20%3D%20%5Cfrac%7B-8.8%7D%20%7B%5Csqrt%7B100.5%7D%7D%20%3D%20-0.878"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>β</mi><mo>=</mo><mfrac><mrow><mover><mi>r</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>r</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>r</mi><mi>y</mi></msub><msqrt><msubsup><mi>r</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>r</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>r</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>8.8</mn></mrow><msqrt><mo stretchy="false">(</mo><mn>2.5</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>8.8</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mn>4.1</mn><msup><mo 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role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> β </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> y </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> r </mi> <mi> y </mi> </msub> <msqrt> <msubsup> <mi> r </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mrow> <mo> − </mo> <mn> 8.8 </mn> </mrow> <msqrt> <mo stretchy="false"> ( </mo> <mn> 2.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 8.8 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 4.1 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mrow> <mo> − </mo> <mn> 8.8 </mn> </mrow> <msqrt> <mn> 100.5 </mn> </msqrt> </mfrac> <mo> = </mo> <mo> − </mo> <mn> 0.878 </mn> </mtd> </mtr> </mtable> </math> </span> </span> </span> <br/> <span style="color: green; "> ::8.8(2.5)2+(-8.8)2+(4.1)2+8.8100.50.878 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-tit"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Cgamma%20%3D%20%5Cfrac%7B%5Coverrightarrow%7Br%7D%20%5Ctimes%20%5Chat%7Bz%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7Br%7D%20%7C%7D%20%3D%20%5Cfrac%7Br_z%7D%20%7B%5Csqrt%7Br%5E2_x%20%2B%20r%5E2_y%20%2B%20r%5E2_z%7D%7D"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>γ</mi><mo>=</mo><mfrac><mrow><mover><mi>r</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>z</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>r</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>r</mi><mi>z</mi></msub><msqrt><msubsup><mi>r</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>r</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>r</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac></mtd></mtr></mtable></math>' id="MathJax-Element-63-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="8.515ex" role="img" style="vertical-align: -3.683ex;" viewbox="0 -2080.5 14890.6 3666.2" width="34.585ex" xmlns:xlink="http://www.w3.org/1999/xlink"> <g fill="currentColor" role="group" stroke="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"> <g role="group" transform="translate(167,0)"> <g role="group" transform="translate(-16,0)"> <g role="group" transform="translate(0,303)"> <use xlink:href="#MJMAIN-63"> </use> <use x="444" xlink:href="#MJMAIN-6F" y="0"> </use> <use x="945" xlink:href="#MJMAIN-73" y="0"> </use> <use x="1589" xlink:href="#MJMATHI-3B3" y="0"> </use> <use x="2410" xlink:href="#MJMAIN-3D" y="0"> </use> <g role="group" transform="translate(3189,0)"> <g role="group" transform="translate(397,0)"> <rect height="60" stroke="none" width="2957" x="0" y="220"> </rect> <g role="group" transform="translate(60,676)"> <use x="274" xlink:href="#MJMATHI-72" y="0"> </use> <use x="55" xlink:href="#MJMAIN-2192" y="530"> </use> <use x="1278" xlink:href="#MJMAIN-D7" y="0"> </use> <g role="group" transform="translate(2279,0)"> <use x="16" xlink:href="#MJMATHI-7A" y="0"> </use> <use x="57" xlink:href="#MJMAIN-5E" y="-13"> </use> </g> </g> <g role="group" transform="translate(672,-1006)"> <use x="0" xlink:href="#MJMAIN-7C" y="0"> </use> <g role="group" transform="translate(278,0)"> <use x="274" xlink:href="#MJMATHI-72" y="0"> </use> <use x="55" xlink:href="#MJMAIN-2192" y="530"> </use> </g> <use x="1334" xlink:href="#MJMAIN-7C" y="0"> </use> </g> </g> </g> <use x="6941" xlink:href="#MJMAIN-3D" y="0"> </use> <g role="group" transform="translate(7720,0)"> <g role="group" transform="translate(397,0)"> <rect height="60" stroke="none" width="6333" x="0" y="220"> </rect> <g role="group" transform="translate(2725,676)"> <use x="0" xlink:href="#MJMATHI-72" y="0"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMATHI-7A" y="-213"> </use> </g> <g role="group" transform="translate(60,-1209)"> <use x="0" xlink:href="#MJSZ2-221A" y="33"> </use> <rect height="60" stroke="none" width="5213" x="1000" y="1124"> </rect> <g role="group" transform="translate(1000,0)"> <use x="0" xlink:href="#MJMATHI-72" y="0"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMAIN-32" y="488"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMATHI-78" y="-211"> </use> <use x="1178" xlink:href="#MJMAIN-2B" y="0"> </use> <g role="group" transform="translate(2179,0)"> <use x="0" xlink:href="#MJMATHI-72" y="0"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMAIN-32" y="488"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMATHI-79" y="-212"> </use> </g> <use x="3306" xlink:href="#MJMAIN-2B" y="0"> </use> <g role="group" transform="translate(4307,0)"> <use x="0" xlink:href="#MJMATHI-72" y="0"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMAIN-32" y="488"> </use> <use transform="scale(0.707)" x="638" xlink:href="#MJMATHI-7A" y="-211"> </use> </g> </g> </g> </g> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> γ </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> z </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> r </mi> <mi> z </mi> </msub> <msqrt> <msubsup> <mi> r </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> </mtd> </mtr> </mtable> </math> </span> </span> </span> <br/> <span style="color: green; "> ::+ry2+rz2+rz2+rz2+ry2+rz2+ry2+rz2+rz2+rz2+rz2+rz2+rz2+ry2+ry2+ry2+ry2+ry2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+ry2+ry2+ry2+ry2+ry2+ry2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz2+rz </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-pfr"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Cgamma%20%3D%20%5Cfrac%7B%5Coverrightarrow%7Br%7D%20%5Ctimes%20%5Chat%7Bz%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7Br%7D%20%7C%7D%20%3D%20%5Cfrac%7Br_z%7D%20%7B%5Csqrt%7Br%5E2_x%20%2B%20r%5E2_y%20%2B%20r%5E2_z%7D%7D%20%3D%20%5Cfrac%7B4.1%7D%20%7B%5Csqrt%7B(2.5)%5E2%20%2B%20(-8.8)%5E2%20%2B%20(4.1)%5E2%7D%7D%20%3D%20%5Cfrac%7B4.1%7D%20%7B%5Csqrt%7B100.5%7D%7D%20%3D%200.409"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>γ</mi><mo>=</mo><mfrac><mrow><mover><mi>r</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>z</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>r</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>r</mi><mi>z</mi></msub><msqrt><msubsup><mi>r</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>r</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>r</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac><mo>=</mo><mfrac><mn>4.1</mn><msqrt><mo stretchy="false">(</mo><mn>2.5</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>8.8</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mn>4.1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mfrac><mn>4.1</mn><msqrt><mn>100.5</mn></msqrt></mfrac><mo>=</mo><mn>0.409</mn></mtd></mtr></mtable></math>' id="MathJax-Element-64-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="8.515ex" role="img" style="vertical-align: -3.683ex;" viewbox="0 -2080.5 36769.4 3666.2" width="85.4ex" xmlns:xlink="http://www.w3.org/1999/xlink"> <g fill="currentColor" role="group" stroke="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"> <g role="group" transform="translate(167,0)"> <g role="group" transform="translate(-16,0)"> <g role="group" transform="translate(0,303)"> <use xlink:href="#MJMAIN-63"> </use> <use x="444" xlink:href="#MJMAIN-6F" y="0"> </use> <use x="945" xlink:href="#MJMAIN-73" y="0"> </use> <use x="1589" xlink:href="#MJMATHI-3B3" y="0"> </use> <use 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height="60" stroke="none" width="2280" x="833" y="785"> </rect> <g role="group" transform="translate(833,0)"> <use xlink:href="#MJMAIN-31"> </use> <use x="500" xlink:href="#MJMAIN-30" y="0"> </use> <use x="1001" xlink:href="#MJMAIN-30" y="0"> </use> <use x="1501" xlink:href="#MJMAIN-2E" y="0"> </use> <use x="1780" xlink:href="#MJMAIN-35" y="0"> </use> </g> </g> </g> </g> <use x="33113" xlink:href="#MJMAIN-3D" y="0"> </use> <g role="group" transform="translate(34170,0)"> <use xlink:href="#MJMAIN-30"> </use> <use x="500" xlink:href="#MJMAIN-2E" y="0"> </use> <use x="779" xlink:href="#MJMAIN-34" y="0"> </use> <use x="1279" xlink:href="#MJMAIN-30" y="0"> </use> <use x="1780" xlink:href="#MJMAIN-39" y="0"> </use> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> γ </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> z </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> r </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> r </mi> <mi> z </mi> </msub> <msqrt> <msubsup> <mi> r </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> r </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mn> 4.1 </mn> <msqrt> <mo stretchy="false"> ( </mo> <mn> 2.5 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 8.8 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> <mo> + </mo> <mo stretchy="false"> ( </mo> <mn> 4.1 </mn> <msup> <mo stretchy="false"> ) </mo> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> = </mo> <mfrac> <mn> 4.1 </mn> <msqrt> <mn> 100.5 </mn> </msqrt> </mfrac> <mo> = </mo> <mn> 0.409 </mn> </mtd> </mtr> </mtable> </math> </span> </span> </span> <br/> <span style="color: green; "> ::+rz2+ry2+rz2=4.1(2.5)2+(-8.8)2+(4.1)2=4.1100.5=0.409 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <h4 id="x-ck12-Y2U0MDdkMmJlMWYyNzA3ZGE1Njc5NGRhODVmZDdjOTc.-260"> Example 8 <br/> <span style="color: green; "> ::例8 </span> </h4> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-YWQxYzU2NzI4ZjZiY2M2NmNmZmQ0OThlNDk3M2E3MDg.-ua7"> Use the method of direction cosines to identify the unit vector having the same direction as the position vector <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Coverrightarrow%7BR%7D%20%3D%20%5Cleft%20%5Clangle%20791%2C%20978%2C%201310%5Cright%20%5Crangle"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mover><mi>R</mi><mo>→</mo></mover><mo>=</mo><mrow><mo>⟨</mo><mrow><mn>791</mn><mo>,</mo><mn>978</mn><mo>,</mo><mn>1310</mn></mrow><mo>⟩</mo></mrow></mtd></mtr></mtable></math>' id="MathJax-Element-65-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="3.917ex" role="img" style="vertical-align: -1.384ex;" viewbox="0 -1090.8 9411 1686.6" width="21.858ex" xmlns:xlink="http://www.w3.org/1999/xlink"> <g fill="currentColor" role="group" stroke="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"> <g role="group" transform="translate(167,0)"> <g role="group" transform="translate(-16,0)"> <g role="group" transform="translate(0,-266)"> <use x="120" xlink:href="#MJMATHI-52" y="0"> </use> <use x="83" xlink:href="#MJMAIN-2192" y="770"> </use> <use x="1361" xlink:href="#MJMAIN-3D" y="0"> </use> <g role="group" transform="translate(2417,0)"> <use x="0" xlink:href="#MJMAIN-27E8" y="0"> </use> <g role="group" transform="translate(389,0)"> <use xlink:href="#MJMAIN-37"> </use> <use x="500" xlink:href="#MJMAIN-39" y="0"> </use> <use x="1001" xlink:href="#MJMAIN-31" y="0"> </use> <use x="1501" xlink:href="#MJMAIN-2C" y="0"> </use> <g role="group" transform="translate(1946,0)"> <use xlink:href="#MJMAIN-39"> </use> <use x="500" xlink:href="#MJMAIN-37" y="0"> </use> <use x="1001" xlink:href="#MJMAIN-38" y="0"> </use> </g> <use x="3448" xlink:href="#MJMAIN-2C" y="0"> </use> <g role="group" transform="translate(3893,0)"> <use xlink:href="#MJMAIN-31"> </use> <use x="500" xlink:href="#MJMAIN-33" y="0"> </use> <use x="1001" xlink:href="#MJMAIN-31" y="0"> </use> <use x="1501" xlink:href="#MJMAIN-30" y="0"> </use> </g> </g> <use x="6284" xlink:href="#MJMAIN-27E9" y="0"> </use> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mover> <mi> R </mi> <mo> → </mo> </mover> <mo> = </mo> <mrow> <mo> ⟨ </mo> <mrow> <mn> 791 </mn> <mo> , </mo> <mn> 978 </mn> <mo> , </mo> <mn> 1310 </mn> </mrow> <mo> ⟩ </mo> </mrow> </mtd> </mtr> </mtable> </math> </span> </span> </span> . <br/> <span style="color: green; "> ::使用方向余弦系方法确定单位矢量,其方向与位置矢量R791,978,1310相同。 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-YzMxYjY5MjhiZWEwNTcxOWE2ZmY0NzhkMTE2MDMyYWE.-9wc"> The unit vector which has the same direction as this vector has the components <br/> <span style="color: green; "> ::与此矢量具有相同方向的单位矢量含有组件的单位矢量 </span> </p> <button class="play-button btn btn-success" style="float: right;" value="@s"> 播放段落 </button> <p id="x-ck12-ZmQzMTg0NTdmN2QwMjMyOGZhYjQ3ZGVjMDRjYzBkOGI.-fsg"> <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Chat%7Bu%7D%20%3D%20%5Cleft%20%5Clangle%20%5Cmbox%7Bcos%7D%5C%20%5Calpha%2C%20%5Cmbox%7Bcos%7D%5C%20%5Cbeta%2C%20%5Cmbox%7Bcos%7D%5C%20%5Cgamma%5Cright%20%5Crangle"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>u</mi><mo stretchy="false">^</mo></mover></mrow><mo>=</mo><mrow><mo>⟨</mo><mrow><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>α</mi><mo>,</mo><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>β</mi><mo>,</mo><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>γ</mi></mrow><mo>⟩</mo></mrow></mtd></mtr></mtable></math>' id="MathJax-Element-66-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="2.768ex" role="img" style="vertical-align: -0.809ex;" viewbox="0 -843.3 10420.7 1191.7" width="24.203ex" xmlns:xlink="http://www.w3.org/1999/xlink"> <g fill="currentColor" role="group" stroke="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"> <g role="group" transform="translate(167,0)"> <g role="group" transform="translate(-16,0)"> <g role="group" transform="translate(0,-25)"> <use x="0" xlink:href="#MJMATHI-75" y="0"> </use> <use x="63" xlink:href="#MJMAIN-5E" y="-13"> </use> <use x="850" xlink:href="#MJMAIN-3D" y="0"> </use> <g role="group" transform="translate(1906,0)"> <use x="0" xlink:href="#MJMAIN-27E8" y="0"> </use> <g role="group" transform="translate(389,0)"> <use xlink:href="#MJMAIN-63"> </use> <use x="444" xlink:href="#MJMAIN-6F" y="0"> </use> <use x="945" xlink:href="#MJMAIN-73" y="0"> </use> <use x="1589" xlink:href="#MJMATHI-3B1" y="0"> </use> <use x="2230" xlink:href="#MJMAIN-2C" y="0"> </use> <g role="group" transform="translate(2675,0)"> <use xlink:href="#MJMAIN-63"> </use> <use x="444" xlink:href="#MJMAIN-6F" y="0"> </use> <use x="945" xlink:href="#MJMAIN-73" y="0"> </use> </g> <use x="4264" xlink:href="#MJMATHI-3B2" y="0"> </use> <use x="4838" xlink:href="#MJMAIN-2C" y="0"> </use> <g role="group" transform="translate(5283,0)"> <use xlink:href="#MJMAIN-63"> </use> <use x="444" xlink:href="#MJMAIN-6F" y="0"> </use> <use x="945" xlink:href="#MJMAIN-73" y="0"> </use> </g> <use x="6872" xlink:href="#MJMATHI-3B3" y="0"> </use> </g> <use x="7805" xlink:href="#MJMAIN-27E9" y="0"> </use> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> <mo> = </mo> <mrow> <mo> ⟨ </mo> <mrow> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> α </mi> <mo> , </mo> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> β </mi> <mo> , </mo> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> γ </mi> </mrow> <mo> ⟩ </mo> </mrow> </mtd> </mtr> </mtable> </math> </span> </span> </span> where <span class="x-ck12-mathEditor" data-contenteditable="false" data-edithtml="" data-math-class="x-ck12-math" data-mathmethod="inline" data-tex="%5Cmbox%7Bcos%7D%5C%20%5Calpha%20%3D%20%5Cfrac%7B%5Coverrightarrow%7BR%7D%20%5Ctimes%20%5Chat%7Bx%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7BR%7D%20%7C%7D%20%3D%20%5Cfrac%7BR_x%7D%20%7B%5Csqrt%7BR%5E2_x%20%2B%20R%5E2_y%20%2B%20R%5E2_z%7D%7D%2C%20%5Cmbox%7Bcos%7D%5C%20%5Cbeta%20%3D%20%5Cfrac%7B%5Coverrightarrow%7BR%7D%20%5Ctimes%20%5Chat%7By%7D%7D%20%7B%20%7C%20%5Coverrightarrow%7BR%7D%20%7C%7D%20%3D%20%5Cfrac%7BR_y%7D%20%7B%5Csqrt%7BR%5E2_x%20%2B%20R%5E2_y%20%2B%20R%5E2_z%7D%7D%2C"> <span class="MathJax_Preview" style="color: inherit; display: none;"> </span> <span class="MathJax_SVG" data-mathml='<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>α</mi><mo>=</mo><mfrac><mrow><mover><mi>R</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>x</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>R</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>R</mi><mi>x</mi></msub><msqrt><msubsup><mi>R</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>R</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>R</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac><mo>,</mo><mstyle displaystyle="false" scriptlevel="0"><mtext>cos</mtext></mstyle><mtext> </mtext><mi>β</mi><mo>=</mo><mfrac><mrow><mover><mi>R</mi><mo>→</mo></mover><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>R</mi><mo>→</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>R</mi><mi>y</mi></msub><msqrt><msubsup><mi>R</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>R</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>R</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></mfrac><mo>,</mo></mtd></mtr></mtable></math>' id="MathJax-Element-67-Frame" role="presentation" style="font-size: 100%; display: inline-block; position: relative;" tabindex="-1"> <svg aria-hidden="true" focusable="false" height="9.205ex" role="img" style="vertical-align: -4.028ex;" viewbox="0 -2229 32233.4 3963.1" width="74.865ex" xmlns:xlink="http://www.w3.org/1999/xlink"> <g fill="currentColor" role="group" stroke="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"> <g role="group" transform="translate(167,0)"> <g role="group" transform="translate(-16,0)"> <g role="group" 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</g> </g> </g> </g> </g> </g> </g> </g> </svg> <span class="MJX_Assistive_MathML" role="presentation"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mtext> cos </mtext> </mstyle> <mtext> </mtext> <mi> γ </mi> <mo> = </mo> <mfrac> <mrow> <mover> <mi> R </mi> <mo> → </mo> </mover> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> z </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mover> <mi> R </mi> <mo> → </mo> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mrow> </mfrac> <mo> = </mo> <mfrac> <msub> <mi> R </mi> <mi> z </mi> </msub> <msqrt> <msubsup> <mi> R </mi> <mi> x </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> R </mi> <mi> y </mi> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> R </mi> <mi> z </mi> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> </mtd> </mtr> </mtable> </math> </span> </span> </span> . <br/> <span style="color: green; "> ::{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{}}}}}}}}{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{}}}}}}}}{{{{{{{{{{{{{{{{{{{{}}}}}}}{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{}}}}}}}}}}}}{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{}}}}}}}}}}}}}}}}}{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{}}}}}}}}}{{{{{{{{}}}}}}}}}}}}}}}{{{{{{{{{{{{{{{{{{{{{}}}}}}}}}}}}}}}}}}}}}}}}}{{{{{{{{{{{{{{}}}}}}}}}}}}}}}}}}}}}}}}}}}}{{{{{{{{{{{{{{{{{{{{{{{{{}}}}}}}}}}}}}}}}}}}}}}}}}}}}}{{{{{{{{{{}}}}}}}}}}}}}}}}}}}}}}}}}{{{{{{{{{{{{{{{{{{{{{}}}}}}}}}}}}}}}}}}}

Once we find the three direction cosines, we have the components of the unit vector,
::一旦我们找到了三个方向的顺带带就有了单位向量的元件

$\begin{aligned} cos α = \frac{\vec{R} \times \hat{x}}{| \vec{R} |} = \frac{791}{\sqrt{791^{2} + 978^{2} + 1310^{2}}} = \frac{791}{\sqrt{329826}} = \frac{791}{1816.11} = 0.436 \end{aligned}$
::=791329826=7911816.11=0.436

$\begin{aligned} cos β = \frac{\vec{R} \times \hat{y}}{| \vec{R} |} = \frac{978}{\sqrt{791^{2} + 978^{2} + 1310^{2}}} = \frac{978}{\sqrt{3298265}} = \frac{978}{1816.11} = 0.539 \end{aligned}$
::=9782+13102=9783298265=9781816.11=0.539

$\begin{aligned} cos γ = \frac{\vec{R} \times \hat{z}}{| \vec{R} |} = \frac{1310}{\sqrt{791^{2} + 978^{2} + 1310^{2}}} = \frac{1310}{\sqrt{3298265}} = \frac{1310}{1816.11} = 0.721 \end{aligned}$
::=131098265=13101816.11=0.721

$\begin{aligned} \hat{u} = ⟨ cos α, cos β, cos γ ⟩ = ⟨ 0.436, 0.539, 0.721 ⟩ \end{aligned}$
::======================================================================================================================================================================================================= =======================================================================================================================================================================================================================================================================

Example 9
::例9

Determine the direction angles between the vector $\begin{aligned} \vec{p} = ⟨ 25, 8, 15 ⟩ \end{aligned}$ and the coordinate axes.
::确定矢量p25,8,15与坐标轴之间的方向角度。

$\begin{aligned} cos α = \frac{\vec{P} \times \hat{x}}{| \vec{P} |} = \frac{P_{x}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::QPxPPPxPx2+Py2+Pz2

$\begin{aligned} cos α = \frac{\vec{P} \times \hat{x}}{| \vec{P} |} = \frac{25}{\sqrt{(25)^{2} + (8)^{2} + (15)^{2}}} = \frac{25}{\sqrt{914}} = \frac{25}{30.23} = 0.827 \end{aligned}$
::+(8)2+(8)2+(15)2=255914=2530.23=0.827

$\begin{aligned} α = {cos}^{- 1} 0.827 = {34.2}^{\circ} \end{aligned}$
::cos- 10.827=34.2

$\begin{aligned} cos β = \frac{\vec{P} \times \hat{y}}{| \vec{P} |} = \frac{P_{y}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::COs PPPPPPyPx2+Py2+Pz2+Pz2+Py2+Py2+Py2+Py2+Py2+PyPyPyPx2+

$\begin{aligned} cos β = \frac{\vec{P} \times \hat{y}}{| \vec{P} |} = \frac{8}{\sqrt{(25)^{2} + (8)^{2} + (15)^{2}}} = \frac{8}{\sqrt{914}} = \frac{8}{30.23} = 0.265 \end{aligned}$
::=8914=830.23=0.265

$\begin{aligned} β = {cos}^{- 1} 0.265 = {75.7}^{\circ} \end{aligned}$
::cos- 10.265=75.7

$\begin{aligned} cos γ = \frac{\vec{P} \times \hat{z}}{| \vec{P} |} = \frac{P_{z}}{\sqrt{P_{x}^{2} + P_{y}^{2} + P_{z}^{2}}} \end{aligned}$
::PzPx2+Py2+Pz2

$\begin{aligned} cos γ = \frac{\vec{P} \times \hat{z}}{| \vec{P} |} = \frac{15}{\sqrt{(25)^{2} + (8)^{2} + (15)^{2}}} = \frac{15}{\sqrt{914}} = \frac{15}{30.23} = 0.496 \end{aligned}$
::15(25)2+(8)2+(15)2+(15)2=15914=1530.23=0.496

$\begin{aligned} γ = {cos}^{- 1} 0.496 = {60.25}^{\circ} \end{aligned}$
::cos-10.496=60.25

Review
::回顾

What is the vector direction in degrees of the following two dimensional vectors, assuming the positive x-axis is 0 ^o ?
::假设正X轴为0,以下两个维矢量度的矢量方向是多少?

What is the direction of $\begin{aligned} ⟨ - 4, 8 ⟩ \end{aligned}$
::4,8的方向是什么?

What is the direction of $\begin{aligned} ⟨ - 9, 20 ⟩ \end{aligned}$
::九点二十分方向何在?

What is the direction of $\begin{aligned} ⟨ 9, 20 ⟩ \end{aligned}$
::九点二十分方向何在?

What is the direction of $\begin{aligned} ⟨ 2, 18 ⟩ \end{aligned}$
::2,18"的方向是什么? 2,18"的方向是什么?

What is the direction of $\begin{aligned} ⟨ 7, 5 ⟩ \end{aligned}$
::7,5的方向是什么?

What is the direction of $\begin{aligned} ⟨ 9, 16 ⟩ \end{aligned}$
::$$9,16$是什么方向? $9,16$

Identify the direction cosines associated with the given vector.
::确定与给定矢量相关的方向余弦。

$\begin{aligned} \vec{P} = ⟨ 42, 6, 9.5 ⟩ \end{aligned}$ a) cos α = b) cos β = c) cos γ =
::P42,6,9.5a) α = b) β = c) cos =

$\begin{aligned} \vec{P} = ⟨ 50, 70, 40.25 ⟩ \end{aligned}$ a) cos α = b) cos β = c) cos γ =
::P50,70,40.25a) α = b) β = c) cos =

$\begin{aligned} \vec{P} = ⟨ 75, 30, 102 ⟩ \end{aligned}$ a) cos α = b) cos β = c) cos γ =
::P 75,30,102 a) cas α = b) cos β = c) cos = cos =

$\begin{aligned} \vec{P} = ⟨ 145, 130, 25.75 ⟩ \end{aligned}$ a) cos α = b) cos β = c) cos γ =
::P145,130,25.75 a) cas α = b) cos β = c) cos = cos =

$\begin{aligned} \vec{P} = ⟨ 220, 300, 175 ⟩ \end{aligned}$ a) cos α = b) cos β = c) cos γ =
::P220 300 175 a) α = b) β = c) cos = γ =

Determine the direction angles between the given vector and the coordinate axes.
::确定给定矢量和坐标轴之间的方向角度。

$\begin{aligned} \vec{P} = ⟨ 13, 30, 17 ⟩ \end{aligned}$
::P 13,30,17

$\begin{aligned} \vec{P} = ⟨ 5, 3, 12 ⟩ \end{aligned}$
::P5,3,12

$\begin{aligned} \vec{P} = ⟨ 75, 130, 45 ⟩ \end{aligned}$
::P75,130,45

$\begin{aligned} \vec{P} = ⟨ 90, 30, 60 ⟩ \end{aligned}$
::P$90,30,60

$\begin{aligned} \vec{P} = ⟨ 7, 18, 4 ⟩ \end{aligned}$
::P7,18,4

Solve the word problems as specified.
::按指定的方式解决字词问题。

Determine the position vector for a kite at the moment it is 3.5 m east, 10.8 m south, and 30 m above the child flying it. Use a coordinate system where the x-direction corresponds to east, the y-direction to north, and the z-direction to the zenith. Then determine the directional cosines to identify the kites location.
::确定风筝在东面3.5米、南面10.8米和上方30米飞行时的位置矢量。使用一个坐标系统,使X方向对准东面,Y方向对准北面,Z方向对准天顶。然后确定方向余弦以识别风筝的位置。

Determine the position vector for a cat stuck in a tree at the moment it is 7 ft. east, 12.3 ft south, and 25 ft. above the base of the tree. Use a coordinate system where the x-direction corresponds to east, the y-direction to north, and the z-direction to the zenith. Then determine the directional cosines to identify the cats location.
::确定被困在树底上方的猫在树上的方位矢量,当时该方位是东经7英尺、南经12.3英尺和树底上方25英尺。使用一个坐标系统,使 x 方向与东对齐,Y 方向与北对齐,Z 方向与天顶对齐。然后确定方向余弦以识别猫的位置。

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

章节大纲

常规

矢向方向

Vector Direction ::矢向方向

Pythagorean Property of Direction Cosines ::方向脊椎动物财产

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Example 9 ::例9

Review ::回顾

Review (Answers) ::回顾(答复)

Vector Direction
::矢向方向

Pythagorean Property of Direction Cosines
::方向脊椎动物财产

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Example 9
::例9

Review
::回顾

Review (Answers)
::回顾(答复)