5.13 直线线段

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  While babysitting your nephew one day you have a discussion regarding math. As a second grader, he is learning to add and subtract quantities. He asks you about what you're doing in math class, and you explain to him that you have just been introduced to vectors. Once you explain to him that a vector is a mathematical quantity that has both magnitude and direction, his question is both simple and yet brilliant: "Why do you need vectors? Can't everything just be described with numbers that don't have direction?"
  ::当有一天你照顾你的侄子的时候,你有一个关于数学的讨论。作为二年级学生,他正在学习增减数量。他问你数学课上你在做什么,你向他解释一下你刚刚被引入向量。当你向他解释向量是一个具有规模和方向的数学数量时,他的问题既简单又聪明:“你为什么需要向量?难道不能用没有方向的数字来描述一切吗?”

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  Directed Line Segments
  ::直线线段

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  A vector is represented diagrammatically by a directed line segment or arrow. A directed line segment has both magnitude and direction . Magnitude refers to the length of the directed line segment and is usually based on a scale. The vector quantity represented, such as influence of the wind or water current may be completely invisible.
  ::矢量由定向线段或箭头以图表形式表示。定向线段既具有大小,也具有方向。磁度指定向线段的长度,通常以比例为基础。表示的矢量,如风或水流的影响,可能是完全看不见的。

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  A 25 mph wind is blowing from the northwest. If $\begin{align*}1\ cm = 5\ mph\end{align*}$ , then the vector would look like this:
  ::从西北部吹出25厘米的风。如果1厘米=5厘米,则矢量会像这样:

  

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  An object affected by this wind would travel in a southeast direction at 25 mph.
  ::受此风影响的物体将朝东南方向25海里处飞行。

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  A vector is said to be in standard position if its initial point is at the origin. The initial point is where the vector begins and the terminal point is where it ends. The axes are arbitrary. They just give a place to draw the vector.
  ::如果矢量的初始点位于源头,则该矢量据说处于标准位置。初始点是矢量的起始点,终点是它的终点。轴是任意的。它们只是为绘制矢量提供一个位置。

  

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  If we know the coordinates of a vector’s initial point and terminal point, we can use these coordinates to find the magnitude and direction of the vector.
  ::如果我们知道矢量初始点和终点的坐标, 我们可以使用这些坐标来找到矢量的大小和方向。

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  All vectors have magnitude . This measures the total distance moved, total velocity, force or acceleration . “Distance” here applies to the magnitude of the vector even though the vector is a measure of velocity, force, or acceleration. In order to find the magnitude of a vector, we use the distance formula. A vector can have a negative magnitude. A force acting on a block pushing it at 20 lbs north can be also written as vector acting on the block from the south with a magnitude of -20 lbs. Such negative magnitudes can be confusing; making a diagram helps. The -20 lbs south can be re-written as +20 lbs north without changing the vector. Magnitude is also called the absolute value of a vector.
  ::所有矢量都有大小。 这可以测量移动的总距离、 全部速度、 力或加速度。 “ 偏差” 适用于矢量的大小, 即使矢量是速度、 力或加速度的量度。 为了找到矢量的大小, 我们使用距离公式。 矢量可以具有负值。 将矢量推向北纬20 lbs的区块上的一股力也可以写成向南区块上以 - 20 lbs 的尺寸作用的矢量。 这种负值可能会令人困惑; 绘制图表会有所帮助。 南部的 - 20 lbs 可以在不改变矢量的情况下重新写成为 + 20 lbs 北面。 磁度也可以称为矢量的绝对值 。

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  Finding the Length of a Vector
  ::查找矢量的长度

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  If a vector starts at the origin and has a terminal point with coordinates (3,5), find the length of the vector.
  ::如果矢量从源点开始,并有一个带有坐标( 3, 5) 的终点, 请找到矢量的长度 。

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  If we know the coordinates of the initial point and the terminal point, we can find the magnitude by using the distance formula. Initial point (0,0) and terminal point (3,5).
  ::如果我们知道初始点和终点的坐标, 我们就可以使用距离公式来找到星号。 初始点( 0, 0) 和终点( 3, 5) 。

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    $\begin{align*}|\vec{v}| = \sqrt{(3 - 0)^2 + (5 - 0)^2} = \sqrt{9 + 25} = 5.8\end{align*}$ The magnitude of $\begin{align*}\vec{v}\end{align*}$ is 5.8.
  :3-0)2+(5-0)2+(5-0)2}9+25=5.8

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  If we don’t know the coordinates of the vector, we must use a ruler and the given scale to find the magnitude. Also notice the notation of a vector, which is usually a lower case letter (typically $\begin{align*}u, v\end{align*}$ , or $\begin{align*}w\end{align*}$ ) in italics, with an arrow over it, which indicates direction. If a vector is in standard position, we can use trigonometric ratios such as sine, cosine and tangent to find the direction of that vector.
  ::如果我们不知道矢量的坐标, 我们就必须使用标尺和给定的尺度来查找星度。 同时注意矢量的标记, 通常用斜体表示一个小写字母( 通常为 u, v 或 w ) , 上面有箭头, 表示方向。 如果矢量处于标准位置, 我们可以使用弦、 弦和正切等三角比来找到该矢量的方向 。

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  Find the Direction of a Vector
  ::查找矢量方向

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  If a vector is in standard position and its terminal point has coordinates of (12, 9) what is the direction?
  ::如果矢量处于标准位置,其终点点的坐标(12,9)是(12,9),那么方向是什么?

  

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  The horizontal distance is 12 while the vertical distance is 9. We can use the tangent function since we know the opposite and adjacent sides of our triangle.
  ::水平距离是12,垂直距离是9,我们可以使用正切函数,因为我们知道三角形的对面和相邻两边。

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  $$\begin{align*}\tan \theta & = \frac{9}{12} \\ \tan^{-1} \frac{9}{12} & = 36.9^\circ\end{align*}$$
  ::912-1912=36.9

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  So, the direction of the vector is $\begin{align*}36.9^\circ\end{align*}$ .
  ::矢量的方向是36.9

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  If the vector isn’t in standard position and we don’t know the coordinates of the terminal point, we must use a protractor to find the direction.
  ::如果矢量不处于标准位置, 我们不知道终点点的坐标, 我们必须使用减速器找到方向 。

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  Two vectors are equal if they have the same magnitude and direction. Look at the figures below for a visual understanding of equal vectors .
  ::如果两个矢量具有相同的大小和方向, 则两个矢量是相等的。 查看下面的数字, 以直观了解相等的矢量 。

  

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  Determin ing if Two Vectors are Equal 
  ::确定两个矢量是否相等

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  $\begin{align*}\vec{a}\end{align*}$ is in standard position with terminal point (-4, 12)
  ::a 处于标准位置,设有终点点(4个、12个)

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  $\begin{align*}\vec{b}\end{align*}$ has an initial point of (7, -6) and terminal point (3, 6)
  ::b 有初始点(7,6)和终点点(3,6)

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  You need to determine if both the magnitude and the direction are the same.
  ::您需要确定大小和方向是否相同。

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  $$\begin{align*}\text{Magnitude}:\quad |\vec{a}| & = \sqrt{(0-(-4))^2 + (0 - 12)^2} = \sqrt{16+144} = \sqrt{160} = 4\sqrt{10} \\ |\vec{b}| & = \sqrt{(7-3)^2 + (-6-6)^2} = \sqrt{16+144} = \sqrt{160} = 4 \sqrt{10} \\ \text{Direction}:\quad \vec{a} & \rightarrow \tan \theta = \frac{12}{-4} \rightarrow \theta = 108.43^\circ \\ \vec{b} & \rightarrow \tan \theta = \frac{-6-6}{7-3} = \frac{-12}{4} \rightarrow \theta = 108.43^\circ\end{align*}$$
  ::磁度: @a_(-0-(-4))2+(0-(-))2+(0-)2+(16)+(144)_(160)=4(10)_(b)+(7-)_(7-6)2+(6-6)_(16)+(160)_(4)_(10)_(12)_(4)_(108)_(43)_(b)_(tan)_(6)_(67)_(3)_(124)_(10843)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  Because the magnitude and the direction are the same, we can conclude that the two vectors are equal.
  ::因为量度和方向相同,我们可以得出结论,这两个矢量是相等的。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to answer your nephew about vectors. 
  ::早些时候,你被要求回答 你侄子关于病媒的问题。

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  Your nephew's thinking is quite good. Many things in the world can be described by numbers, without the use of direction. However, math needs to "line up" with reality. If something doesn't work with just numbers, there needs to be a new type of mathematical quantity to describe the behavior completely. For example, consider two cars that are both moving at 25 miles per hour. Will they collide?
  ::你侄子的想法相当不错。 世界上有许多事情可以用数字来描述, 不使用方向。 然而, 数学需要与现实“ 连接起来 ” 。 如果数字不起作用, 则需要有一种新型的数学数量来完全描述行为。 比如, 考虑两辆汽车, 它们每小时以25英里的速度移动。 它们会相撞吗 ?

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  You can see that there isn't enough information to answer the question. You don't know which way the cars are going. If the two cars are going in the same direction, then they won't collide. If, however, they are going directly at each other, then they will certainly collide. In order to describe the behavior of the cars completely, a quantity is needed that is not just the magnitude of the car's motion, but also the direction - which is why vectors are needed.
  ::你可以看到没有足够的信息来解答问题。 您不知道汽车往哪个方向行驶。 如果两辆汽车向同一方向行驶, 那么他们就不会相撞。 但是, 如果两辆汽车直接向对方行驶, 那么他们肯定会相撞。 为了完整描述汽车的行为, 不仅需要数量, 不仅仅是汽车运动的大小, 还需要方向, 这也是需要矢量的原因。

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  And that is how you should answer your nephew.
  ::你该这样回答你的侄子

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  Example 2
  ::例2

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  Given the initial and terminal coordinates below, find the magnitude and direction of the vector that results.
  ::考虑到下面的初始坐标和终端坐标, 找到结果的矢量的大小和方向 。

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  initial ( 2, 4) terminal (8, 6)
  ::终端(8,6)

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  $\begin{align*}|\vec{a}| = \sqrt{(2-8)^2 + (4-6)^2} = 6.3\end{align*}$ , direction $\begin{align*}=\tan^{-1} \left (\frac{4-6}{2-8} \right ) = 18.4^\circ\end{align*}$
  ::a(2-8)2+(4-6)2=6.3,方向=tan-1(4-62-8)=18.4

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  Example 3
  ::例3

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  Given the initial and terminal coordinates below, find the magnitude and direction of the vector that results.
  ::考虑到下面的初始坐标和终端坐标, 找到结果的矢量的大小和方向 。

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  initial (5, -2) terminal (3, 1)
  ::终端(3,1)

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    $\begin{align*}|\vec{a}| = \sqrt{(5-3)^2 + (-2-1)^2} = 3.6\end{align*}$ , direction $\begin{align*}=\tan^{-1} \left (\frac{-2-1}{5-3} \right ) = 123.7^\circ\end{align*}$ . Note that when you use your calculator to solve for $\begin{align*}\tan^{-1} (\frac{-2-1}{5-3})\end{align*}$ , you will get $\begin{align*}-56.3^\circ\end{align*}$ . The calculator produces this answer because the range of the calculator’s $\begin{align*}y = \tan^{-1} x\end{align*}$ function is limited to $\begin{align*}-90^\circ < y < 90^\circ\end{align*}$ . You need to sketch a draft of the vector to see that its direction when placed in standard position is into the second quadrant (and not the fourth quadrant), and so the correct angle is calculated by moving the angle into the second quadrant through the equation $\begin{align*}-56.3^\circ + 180^\circ=123.7^\circ.\end{align*}$
  :5-3) 2+ (-2- 1) 2=3.6, 方向=tan-1 (-2-15-3) =123.7 。 注意当您使用计算器解析 tan-1 (-2- -15-3) 时, 您将得到 -56. 3 。 计算器生成这个答案, 因为计算器 y y =- 1x 函数的范围限于 - 90 y < 90 。 您需要绘制矢量的草稿, 才能看到将矢量置于标准位置时的方向位于第二个象限( 而不是第四个象限) 。 因此, 正确的角度是通过将角度移入第二个象限 等方 -56. 3 180 123. + 。

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  Example 4
  ::例4

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  Assume $\begin{align*}\vec{a}\end{align*}$ is in standard position. For the terminal point (12, 18), find the magnitude and direction of the vector.
  ::Asssume a 处于标准位置。 对于终点( 12, 18) , 找到矢量的大小和方向 。

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  $\begin{align*}|\vec{a}| = \sqrt{12^2 + 18^2} = 21.6\end{align*}$ , direction $\begin{align*}=\tan^{-1} \left (\frac{18}{12} \right ) = 56.3^\circ \end{align*}$
  ::a122+182=21.6,方向=tan-1(1812)=56.3

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  Review
  ::回顾

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  1. What is the difference between the magnitude and direction of a vector?
     ::矢量的大小和方向之间有什么区别?
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  2. How can you determine the magnitude of a vector if you know its initial point and terminal point?
     ::如果您知道矢量的初始点和终点, 您如何确定矢量的大小 ?
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  3. How can you determine the direction of a vector if you know its initial point and terminal point?
     ::如果您知道矢量的初始点和终点, 您如何确定矢量的方向 ?
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  4. How can you determine whether or not two vectors are equal?
     ::您如何确定两个矢量是否相等 ?
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  5. If a vector starts at the origin and has a terminal point with coordinates (2, 7), find the magnitude of the vector.
     ::如果矢量从源点开始,并有一个带有坐标(2、7)的终点,则会发现矢量的大小。
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  6. If a vector is in standard position and its terminal point has coordinates of (3, 9), what is the direction of the vector?
     ::如果矢量处于标准位置,其终点点的坐标为(3,9),则矢量的方向是什么?
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  7. If a vector has an initial point at (1, 6) and has a terminal point at (5, 9), find the magnitude of the vector.
     ::如果矢量的初始点为(1、6)和终点点为(5、9),则查找矢量的大小。
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  8. If a vector has an initial point at (1, 4) and has a terminal point at (8, 7), what is the direction of the vector?
     ::如果矢量的初始点为(1, 4),终点点为( 8, 7),则矢量的方向是什么?

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  Given the initial and terminal coordinates below, find the magnitude and direction of the vector that results.
  ::考虑到下面的初始坐标和终端坐标, 找到结果的矢量的大小和方向 。

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  9. initial (4, -1); terminal (5, 3)
     ::终端(5,3)
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  10. initial (2, -3); terminal (4, 5)
      ::终端 (4,5)
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  11. initial (3, 2); terminal (0, 3)
      ::初始(3,2);终端(0,3)
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  12. initial (-2, 5); terminal (2, 1)
      ::初始 (-2, 5); 终端(2, 1)

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  Determine if the two vectors are equal.
  ::确定两个矢量是否相等。

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  13. $\begin{align*}\vec{a}\end{align*}$ is in standard position with terminal point (1, 5) and $\begin{align*}\vec{b}\end{align*}$ has an initial point (3, -2) and terminal point (4, 2).
      ::a 处于标准位置,终点(1,5),b 初始点(3,2)和终点点(4,2)为初始点(3,2)。
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  14. $\begin{align*}\vec{c}\end{align*}$ has an initial point (-3, 1) and terminal point (1, 2) and $\begin{align*}\vec{d}\end{align*}$ has an initial point (3, 5) and terminal point (7, 6).
      ::c 有初始点(3,1),终点(1,2)和终点(7,6)有初始点(3,5)和终点(7,6)。
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  15. $\begin{align*}\vec{e}\end{align*}$ is in standard position with terminal point (2, 3) and $\begin{align*}\vec{f}\end{align*}$ has an initial point (1, -6) and terminal point (3, -9).
      ::e 处于标准位置,有终点(2、3),f 具有初始点(1、6)和终点点(3、9)。

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  Review (Answers)
  ::回顾(答复)

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  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 。