两维矢量

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Introduction
::导言

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An airplane being pushed off course by wind and a swimmer's movement across a flowing river are both examples of vectors in action. 
::飞机被风推离航向 游泳者穿越流河的移动 都成为了病媒活动的例子

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Points in the coordinate plane describe a location. Vectors, on the other hand, have no location and indicate only magnitude and  direction. Vectors can describe the strength of forces like gravity or speed and the direction of a ship at sea. Vectors are extremely useful in modeling complex situations in the real world. 
::坐标平面上的点描述一个位置。另一方面,向量没有位置,只能表示大小和方向。向量可以描述引力或速度以及海上船只的方向。向量在模拟真实世界的复杂情况方面非常有用。

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What are other differences between points and vectors? 
::点和矢量之间的其他区别是什么?

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Vectors
::矢量

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One way to define a vector is as a line segment with a direction.  Vectors can be represented graphically using an arrow that points from the vector's initial point, called the  tail , to its terminal point, called the  head . 
::定义矢量的一种方式是作为带有方向的线条段。矢量可以用一个箭头用图形表示,箭头指向矢量的初始点,即尾端,到终点,即头部。

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The two defining characteristics of a vector are its magnitude and its direction. The magnitude is shown graphically by the length of the arrow, and the direction is indicated by the angle in which the arrow is pointing. The direction of a vector on the Cartesian coordinate plane is determined by its counter-clockwise rotation from the positive side of the $\begin{align*}x\end{align*}$ -axis. This can be seen in the images below:
::矢量的两个定义特性是其大小和方向。该数值用箭头的长度图形显示,方向则用箭头指向的角表示。笛卡尔坐标平面上的矢量方向由从 X 轴正侧反时针旋转决定。以下图像中可以看到这一点:

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Notice how the vector below is shown multiple times on the same coordinate plane. This image emphasizes that the location on the coordinate plane does not matter and is not unique. Each representation of the vector has identical direction and magnitude.
::注意如何在同一坐标平面上多次显示下面的矢量。 此图像强调坐标平面上的位置无关紧要, 也不是独一无二的。 矢量的每个表达式的方向和大小相同 。

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There are a few different ways to symbolize a vector,  $\begin{align*}v\end{align*}$ :
::有几种不同的方式来象征矢量,v:

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$$\begin{align*}v, \ \overrightarrow{v}, \ \text{or} \ \overset{\rightharpoonup}{v.}\end{align*}$$
::v、v________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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There are also a few ways to describe a specific vector. First, you can describe its angle and magnitude. Second, you can describe it as an ordered pair or in component form ,  $\begin{align*}\langle x, y \rangle\end{align*}$ .  Note that when discussing vectors ,  you should use brackets $\begin{align*}\langle \ \rangle\end{align*}$  instead of parentheses to help avoid confusion between a vector and a point. Vectors can be two-dimensional, three-dimensional, or $\begin{align*}n\end{align*}$ -dimensional, where $\begin{align*}n\end{align*}$ is an integer greater than 3.
::也可以用几种方法描述特定的矢量。首先,您可以描述其角度和大小。第二,您可以把它描述为定单对或构件形式,x,y。请注意,在讨论矢量时,您应该使用括号 ,而不是括号来帮助避免矢量和点之间的混淆。矢量可以是二维、三维或n维,其中n的整数大于3。矢量可以是二维、三维或n维,其中n的整数大于3。

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The following video discusses vectors and vector vocabulary: 
::以下视频讨论矢量和矢量词汇表:

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When a vector is angled in a certain direction, it is often useful to break the vector into individual components. The  component of a vector depicts the influence of that vector in a given direction. A   two-dimensional vector can be thought of as having an influence in two different directions: the horizontal direction ( $\begin{align*}x\end{align*}$ -axis) and the vertical direction ( $\begin{align*}y\end{align*}$ -axis). 
::当矢量按某一方向角度向上,将矢量折成个别组成部分往往有用。矢量的构成部分描述该矢量在某一方向的影响。二维矢量可被视为对两个不同方向有影响:横向方向(x轴)和垂直方向(y轴)。

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For example, the vector on the left in the image below has components in both the  $\begin{align*}x\end{align*}$ -direction,  AB x , and the $\begin{align*}y\end{align*}$ -direction,  AB y . Here,  AB x  = 4 and  AB y  = 3 because point B is 4 units to the right and 3 units up from point A. Similarly, vector  D  has components  D x  = 3 and  D y  = –1.25. The negative sign in  D y  indicates that the  $\begin{align*}y\end{align*}$  component of vector  D  is downward, in the negativ e  $\begin{align*}y\end{align*}$ -direction . As with other numbers, we usually only include the negative sign explicitly.
::例如,下方图像左侧的矢量在 x 方向、 ABx 和 Y 方向中都有组件。 这里, ABx = 4 和 ABy = 3 , 因为点B 向右是 4 个单位, 从 A 点向上是 3 个单位。 同样, 矢量 D 有 构件 Dx = 3 和 Dy = - 1. 25 。 Dy 中的负符号表示矢量D 的 y 组成部分在负 Y 方向下方。 与其他数字一样, 我们通常只明确包括负符号 。

 

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The following video further  explains how to find the component form of a vector given the vector's magnitude and direction: 
::以下视频进一步解释根据矢量的大小和方向如何找到矢量的构成形式:

 

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To determine  the vector between two points, start with the terminal point and subtract the initial point. Suppose   $\begin{align*}A (u, v) \text{ and } B (x, y)\end{align*}$ . Then:  
::要确定两个点之间的矢量,请从终点开始,减去起始点。假设 A(u,v) 和 B(x,y) 。然后:

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$$\begin{align*}\vec{AB} &= \langle x-u, y-v\rangle \\ \vec{BA} &= \langle u-x, v-y\rangle . \end{align*}$$  
::AB -x -u,y -v -v -BA -u -x,v -y

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Vectors are said to be equal if they have the same magnitude and the same direction.  The absolute value of a vector $\begin{align*}\left| \vec{v} \right|\end{align*}$  is the same as the length or magnitude of the vector .  Magnitude can be found by using the Pythagorean Theorem or the distance formula.   
::如果矢量的大小和方向相同,则矢量是相等的。矢量 v的绝对值与矢量的长度或大小相同。使用 Pythagoren定理或距离公式可以找到磁度。

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Play, Learn, and Explore with two-dimensional vectors using the example of a river ferry and currents: .
::使用河流渡轮和海流的例子使用二维矢量玩、学习和探索: 。

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Examples
::实例

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Example 1
::例1

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Consider the points  $\begin{align*}A (1, 3), B (-4, -6), C (5, -13)\end{align*}$ .  Find the vectors in component form of $\begin{align*}\overrightarrow{AB}, \overrightarrow{BA}, \overrightarrow{AC}, \overrightarrow{CB}\end{align*}$ . 
::考虑A(1,3)、B(-4,6)、C(5,13)点,找出以AB__,BA__,AC__,CB__为组成部分的矢量。

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Solution:
::解决方案 :

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$$\begin{align*}\overrightarrow{AB} &= \langle -5, -9 \rangle \\ \overrightarrow{BA} &= \langle 5, 9 \rangle \\ \overrightarrow{AC} &= \langle 4, -16 \rangle \\ \overrightarrow{CB} &= \langle -9, 7 \rangle \end{align*}$$
::AB5,9-9-BA5,5-9-9-9-9-5-AC4,4-16-CB9,7-

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Example 2
::例2

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What are the $\begin{align*}x\end{align*}$  and $\begin{align*}y\end{align*}$  components of the vectors shown in the diagram?
::图中显示的矢量的 x 和 Y 组件是什么 ?

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Solution:
::解决方案 :

	AB	CD	EF	GH	IJ	KL	MN
$\begin{align*}x\end{align*}$  component	2.25	-1.5	0	-1.5	2.0	1.5	-2.25
$\begin{align*}y\end{align*}$   component	0.25	-2.0	-2.25	-2.0	-1.5	2.0	0

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In the diagram, each division is 0.25 units. All vectors that point toward the left have negative $\begin{align*}x\end{align*}$  components, and those that point downward have negative   $\begin{align*}y\end{align*}$  c omponents. Notice that for the horizontal vector, MN , the $\begin{align*}y\end{align*}$  component is equal to 0. Likewise, for the vertical vector, EF , the $\begin{align*}x\end{align*}$  component is equal to 0.
::在图表中,每个区域为0.25个单位。所有向左点的矢量都有负 x 组件,向下方向的矢量有负 Y 组件。请注意,对于水平矢量, MN, Y 组件等于 0。同样,对于垂直矢量, EF, x 组件等于 0。

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Example 3
::例3

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1) Which of the vectors in Example 2 is equal to vector  CD ?
:1) 例2中的哪个矢量等于矢量 CD?

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Solution:
::解决方案 :

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Vector   GH   =   CD.   Both vectors have the same length and the same direction or orientation.
::矢量 GH = CD。两种矢量的长度相同,方向或方向相同。

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2) Which vector is equal to  –CD ?
:2) 哪个矢量等于-CD?

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Solution:
::解决方案 :

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Vector  KL  =  –CD . Both vectors have the same length, and the two vectors point in opposite directions.
::矢量 KL =- CD。两个矢量的长度相同,而两个矢量的方向相反。

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Example 4

::例4

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Return to the problem from the Introduction: W hat are other differences between points and vectors? 
::回到引言中的问题:点和矢量之间还有什么其他区别?

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Solution:
::解决方案 :

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There are many differences between points and  vectors . Points are specific locations, and vectors are made up of distance and angles. Parentheses are used for points, and $\begin{align*}\langle \ \rangle \end{align*}$  are used for vectors. Without the starting point, the vector could start from anywhere. 
::点和矢量之间有许多差异。 点是特定的位置, 矢量由距离和角度组成。 括号用于点, 用于矢量。 没有起点, 矢量可以从任何地方开始 。

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Example 5
::例5

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A marine traffic controller sees a ship traveling NNW at 17 knots (nautical mph) on radar.  Describe the ship's movement in component form using a vector.
::海洋交通控制器在雷达上看到一艘在17海里处航行的NNW的船舶,用矢量说明该船舶的部件形式移动情况。

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Solution:
::解决方案 :

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The marine traffic controller is using cardinal direction which is based on the orientation of a compass.
::海洋交通管制员正在使用以指南针方向为基础的主方向。

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NNW is halfway between NW and N. When you describe ships at sea, it is best to use a bearing that has $\begin{align*}0^\circ\end{align*}$  as due North and $\begin{align*}270^\circ\end{align*}$  as due West. This makes NW equal to  $\begin{align*}315^\circ\end{align*}$ and NNW equal to $\begin{align*}337.5^\circ\end{align*}$ .
::NNW在NW和N之间是一半。当你描述海上船只时,最好使用北纬0°C和西经270°C的轴承。这使得NW等于315°C,西北等于337.5°C。

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When the problem is depicted above, it becomes a basic trig question to find the $\begin{align*}x\end{align*}$  and $\begin{align*}y\end{align*}$ components of the vector. Note that the reference angle the vector makes with the negative portion of the  $\begin{align*}x\end{align*}$ axis is $\begin{align*}67.5^\circ\end{align*}$ .
::当上述问题被描述时,找到矢量的 x 和 y 组成部分就成为一个基本的三重问题。请注意,矢量与 x 轴负部分的参考角为 67.5 。

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$$\begin{align*}\sin 67.5^\circ &= \frac{y}{17}, \cos 67.5^\circ=\frac{x}{17}\\ < x, y > & \approx < -6.5, 15.7 > \end{align*}$$
::67.5y17,cos67.5x17 <x,y6.5.15.7>

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Note :   $\begin{align*}270^\circ\end{align*}$   clockwise is equivalent to  $\begin{align*}90^\circ\end{align*}$  counterclockwise. This problem can also be worked using this angle measurement.
::注意: 270- 顺时针等于 90/ 逆时针。 使用此角度测量也可以解决这个问题 。

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Example 6

::例6

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A father and daughter are sledding. After going downhill, the father pulls his daughter up the hill using a rope attached to the sled. The sled sits on the ground. The hill has a $\begin{align*}20^\circ\end{align*}$  incline. The rope makes a $\begin{align*}39^\circ\end{align*}$  angle with the slope. Draw a force diagram showing how these forces act on the daughter's center of gravity:
::父亲和女儿在滑雪。在下坡后,父亲用与雪橇相连的绳子拉着女儿上山。 雪橇坐落在地上。 山上有20英寸的刺线。 绳子用斜坡的角度为39英寸的角。 绘制一个显示这些部队如何在女儿的重心上行动的力量图 :

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a. The force of gravity.
::a. 重力。

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b. The force holding the daughter in the sled to the ground.
::b. 将女儿放在雪橇上拖到地上的部队。

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c. The force pulling the daughter backward down the slope.
::c. 将女儿拉下斜坡的部队。

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d. The force of the father pulling the daughter up the slope.
::d. 父亲将女儿拉上斜坡时,父亲的力量。

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Solution:
::解决方案 :

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The girl's center of gravity is represented by the black dot. The force of gravity is the black arrow straight down. The green arrow is the effect of gravity pulling the girl down the slope. The red arrow is the effect of gravity pulling the girl straight into the slope. The blue arrow represents the force the father is exerting as he pulls the girl up the hill.
::女孩的重力中心由黑点代表。 重力的力量是黑箭直下。 绿箭是引力的影响, 将女孩拉下斜坡。 红箭是引力的影响, 将女孩拉进斜坡。 蓝箭代表父亲将女孩拉上山时所施加的力量。

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Notice that the father's force vector (blue) is longer than the force pulling the girl down the hill.  This means that, over time, father and daughter will make progress and ascend the hill. Also, note that the father is wasting some of his energy by lifting rather than just pulling. If he could pull at an angle directly opposing the force pulling the girl down the hill, he would be using all his energy efficiently. 
::请注意,父亲的力量向量(蓝色)比把女孩拉下山的部队长。这意味着,随着时间的推移,父亲和女儿将进步并攀登山丘。 另外,父亲正在浪费一些精力,举起而不是只是拉动。 如果他能直接从一个角度拉动部队把女孩拉下山,他就会高效地利用他的所有能量。

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Example 7

::例7

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Center the force diagram from the previous question onto the origin, and identify the angle between each consecutive force vector. 
::将上一个问题中的强制图居中到源头,并确定每个连续的强制矢量之间的角。

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Solution:
::解决方案 :

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T he $\begin{align*}x\end{align*}$ - and $\begin{align*}y\end{align*}$ -axes are included as a reference; note that the gravity vector overlaps with the negative $\begin{align*}y\end{align*}$  axis. To find each angle, you must use your knowledge of supplementary, complementary, and vertical angles and all the clues from the question. To check, see if all the angles sum to be $\begin{align*}360^\circ\end{align*}$ .
::x 和 y 轴作为参考; 注意重力矢量与负 y 轴相重叠。 要找到每个角度, 您必须使用您对补充、 互补和垂直角度以及所有问题线索的知识。 要检查, 请检查是否所有角度都等于 360 。

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Review
::回顾

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1. Describe what a vector is and give a real-life example of something that a vector could model.
::1. 描述矢量是什么,并举例说明矢量可以模拟的东西的真实生活实例。

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Consider the points  $\begin{align*}A (3, 5), B(-2,-4), C(1,-12), D(-5, 7)\end{align*}$ .  Find the vectors in component form of:
::考虑A(3,5)、B(-2,4),C(1,-12),D(-5,7)点。

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2. $\begin{align*}\overrightarrow{AB}\end{align*}$
::2. AB_____________________________________________________________________________________________________________________

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3.  $\begin{align*}\overrightarrow{BA}\end{align*}$
::3. BA_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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4.  $\begin{align*}\overrightarrow{AC}\end{align*}$
::4. AC__________________________________________________________________________________________________________________

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5.  $\begin{align*}\overrightarrow{CB}\end{align*}$
::5. CB_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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6.  $\begin{align*}\overrightarrow{AD}\end{align*}$
::6. AD____________________________________________________________________________________________

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7. $\begin{align*}\overrightarrow{DA}\end{align*}$
::7. 达 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州 州

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For each of the following vectors, draw the vector on a coordinate plane starting at the origin, and find its magnitude:
::对于下列每种矢量,从源头开始,在坐标平面上绘制矢量,并发现其大小:

8.  $\begin{align*}\langle 3, 7 \rangle \end{align*}$

9.  $\begin{align*}\langle -3, 4 \rangle \end{align*}$

10.  $\begin{align*}\langle -5, 10 \rangle \end{align*}$

11.  $\begin{align*}\langle 6, -8 \rangle \end{align*}$

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12. Can the $\begin{align*}x\end{align*}$  or $\begin{align*}y\end{align*}$  component of a vector ever have a greater magnitude than the vector itself?
::12. 矢量的 x 或 Y 成分是否比矢量本身具有更大的规模?

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13. If two vectors have magnitudes that are not equal, can the sums of their magnitudes ever be zero?
::13. 如果两个矢量的量值不相等,那么其量值的总和能否为零?

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14. A ship is traveling SSW at 13 knots. Describe this ship's movement in a vector. 
::14. 一艘船舶在南南纬13海里处行驶,说明该船舶在矢量中的动向。

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15. A vector that describes a ship's movement is $\begin{align*}\langle 5 \sqrt{2}, 5 \sqrt{2} \rangle \end{align*}$ .  In what direction is the ship traveling, and what is its speed in knots? 
::15. 描述船舶运动的矢量是52,52。船舶驶往哪个方向,其节节速度如何?

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16. If a boat is being motored perpendicularly at 35 km/hr across a stream that is flowing at 25 km/hr, how can the direction and speed it travels be clearly shown using vectors? 
::16. 如果一艘船在每小时35公里/小时的直径35公里/小时的一条溪流中行驶,每小时25公里/小时,如何用矢量清楚地显示其行驶的方向和速度?

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Review (Answers )
::回顾(答复)

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Please see the Appendix.
::请参看附录。