5.18 矢量和斜坡的翻译

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  矢量和斜坡的翻译

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  You and your friends are at a ski weekend for your school. While skiing, you go down a hill that is rather steep. You decide to use a vector to represent your motion from the top of the hill going down. Counting the top of the hill as the origin, you ski down a slope and measure how far your "x" and "y" positions have changed. As it turns out, you can represent this displacement with the vector $\begin{align*}(12, -256)\end{align*}$ . Can you calculate the incline (slope) of the hill you came down?
  ::你和你的朋友们在周末上学滑雪。滑雪时,你们下山时要跳得相当陡峭。你们决定用矢量来代表你们从山顶上往下运动。把山顶计为起源,滑下坡坡,测量你们的“x”和“y”位置变化了多少。结果显示,你们可以用矢量(12,-256)来代表这种迁移(12,2-256),你能计算下山的内线吗?

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  Translating Vectors and Slopes
  ::传导矢量和斯隆

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  Vectors with the same magnitude and direction are equal. This means that the same ordered pair could represent many different vectors.
  ::向量和方向相同的向量相等。 这意味着同一对定购的向量可以代表许多不同的向量。

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  For instance, the ordered pair (4, 8) can represent a vector in standard position where the initial point is at the origin and the terminal point is at (4, 8). This vector could be thought of as the of a horizontal vector with a magnitude or 4 units and a vertical vector with a magnitude of 8 units. Therefore, any vector with a horizontal component of 4 and vertical component of 8 could also be represented by the ordered pair (4, 8).
  ::例如,定单对(4,8)可以代表标准位置的矢量,初始点位于起始点,终点点位于(4,8),该矢量可被视为具有星度或4个单位的水平矢量,垂直矢量为8个单位的垂直矢量,因此,任何具有水平组成部分为4和垂直组成部分为8的矢量也可以由定单对(4,8)代表。

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  If you think back to Algebra, you know that the slope of a line is the change in $\begin{align*}y\end{align*}$ over the change in $\begin{align*}x\end{align*}$ , or the vertical change over the horizontal change.
  ::如果您回想到代数, 你知道线的斜坡是 y 相对于 x 的变化的变化, 或者水平变化的垂直变化 。

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  Let's take a look at some problems that involve translating vectors and slopes. 
  ::让我们来看看有些问题 涉及到传导矢量和斜坡的翻译

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  1. Consider the vector from (4, 7) to (12, 11). What would the representation of a vector that had 2.5 times the magnitude be?
  ::1. 将矢量从(4,7)到(12,11)考虑进去。

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  Here, $\begin{align*}k = 2.5\end{align*}$ and $\begin{align*}\vec{v} =\end{align*}$ the directed segment from (4, 7) to (12, 11).
  ::在这里, k=2.5 和 v 方向段从 (4, 7) 到 (12, 11) 。

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  Mathematically, two vectors are equal if their direction and magnitude are the same. The positions of the vectors do not matter. This means that if we have a vector that is not in standard position, we can translate it to the origin. The initial point of $\begin{align*}\vec{v}\end{align*}$ is (4, 7). In order to translate this to the origin, we would need to add (-4, -7) to both the initial and terminal points of the vector.
  ::从数学角度讲,如果两个矢量的方向和大小相同,则两个矢量是相等的。矢量的位置无关紧要。这意味着如果我们的矢量不处于标准位置,我们可以将其转换为源。 v的起始点是 (4, 7) 。为了将其转换为源,我们需要在矢量的初始点和终端点上添加( 4, 7) 。

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  Initial point: $\begin{align*}(4, 7) + (-4, -7) = (0,0)\end{align*}$
  ::初始点: (4,7)+(-4,7)=(0,0)

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  Terminal point: $\begin{align*}(12, 11) + (-4, -7) = (8, 4)\end{align*}$
  ::终点: (12,11)+(-4,-7)=(8,4)

  

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  Now, to calculate $\begin{align*}\vec{kv}\end{align*}$ :
  ::现在,要计算 kv :

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  $$\begin{align*}\vec{kv} & = (2.5(8), 2.5(4)) \\ \vec{kv} & = (20,10)\end{align*}$$
  ::kv( 2.5(8), 2.5(4))kv( 20, 10)

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  The new coordinates of the directed segment are (0, 0) and (20, 10). To translate this back to our original terminal point:
  ::定向段的新坐标是(0,0)和(20,10)。

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  Initial point: $\begin{align*}(0, 0) + (4, 7) = (4, 7) \end{align*}$
  ::初始点: (0,0)+(4,7)=(4,7)

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  Terminal point: $\begin{align*}(20, 10) + (4, 7) = (24, 17)\end{align*}$
  ::终点: (20,10)+(4,7)=(24,17)

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  The new coordinates of the directed segment are (4, 7) and (24, 17).
  ::指示段的新坐标是(4,7)和(24,17)。

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  2. What is the slope of a vector starting from the origin with terminal coordinates (5 , 7)?
  ::2. 矢量的斜坡是什么,从源头起,带有终端坐标(5,7)?

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  Since the slope is defined as the change in "y" divided by the change in "x", we can find the slope of this vector:
  ::由于斜坡的定义是“y”的变动除以“x”的变动,我们可以找到该矢量的斜坡:

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  $$\begin{align*} \frac{\Delta y}{\Delta x} = \frac{7 - 0}{5 - 0} = \frac{7}{5} = 1.4 \end{align*}$$
  ::yx=7-05-0=75=1.4

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  3. Find the new coordinates of the vectors
  ::3. 寻找矢量的新坐标

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  A vector starts at the origin and has terminal coordinates (11 , 17). What would the new coordinates of the tail and tip of the vector be if the vector were shifted 15 units along the "x" axis?
  ::矢量从源点开始,有终端坐标(11,17)。如果矢量沿“x”轴移动15个单位,则矢量尾部和端部的新坐标是什么?

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  The vector maintains the same orientation in space, it is just moved down the "x" axis. Therefore, only the "x" coordinates of the vector's tail and tip change.
  ::矢量在空间中保持相同的方向, 它只是移到“ x” 轴下。 因此, 只有矢量尾部和尾部变化的“ x” 坐标 。

  

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  So the new coordinates of the tail of the vector are:
  ::因此矢量尾部的新坐标是:

  $$\begin{align*} (0 + 15, 0) = (15,0) \end{align*}$$

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  And the new coordinates of the tip are:
  ::线索的新坐标是:

  $$\begin{align*} (11 + 15, 17) = (26,17) \end{align*}$$

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to calculate the incline (slope) of the hill you came down. 
  ::早些时候,有人要求你计算你所下山的斜坡。

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  Since the slope is defined as the change in "y" divided by the change in "x", we can find the slope of the vector representing your trip down the hill:
  ::由于斜坡的定义是“y”的变动除以“x”的变动,我们可以找到矢量的斜坡,表示您在山上旅行:

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  $$\begin{align*} \frac{\Delta y}{\Delta x} = \frac{-256 - 0}{12 - 0} = \frac{-256}{12} \approx -21.33 \end{align*}$$
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}...

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  This means that for every foot the hill changed in the "x" direction, it went down 21.33 feet in the "y" direction. That's a steep hill indeed!
  ::也就是说,对于每一英尺的山体, 山体在“x”方向上的变化, 山体在“y”方向下方21.33英尺。这的确是一个陡峭的山体!

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  Example 2
  ::例2

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  Find the magnitude of the horizontal and vertical components of the following vector given the following coordinates of their initial and terminal points.
  ::根据下列初始点和终点点的坐标,查找下列矢量的横向和垂直组成部分的大小。

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  $\begin{align*}\text{initial} = (-3,8) \qquad \qquad \quad \text{terminal} = (2, -1)\end{align*}$
  ::初始=( - 3,8) 终点=( 2, - 1)

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  The vector needs to be translated to (0,0). Also, recall that magnitudes are always positive.
  ::矢量需要转换为(0,0),还要提醒注意,数值总是正值。

  $\begin{align*}(-3, 8) + (3, -8) = (0, 0) \qquad \qquad (2, -1) + (3, -8) = (5, -9)\end{align*}$

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  horizontal $\begin{align*}= 5\end{align*}$ , vertical $\begin{align*}= 9\end{align*}$
  ::水平=5,垂直=9

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  Example 3
  ::例3

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  Find the magnitude of the horizontal and vertical components of the following vector given the following coordinates of their initial and terminal points.
  ::根据下列初始点和终点点的坐标,查找下列矢量的横向和垂直组成部分的大小。

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  $\begin{align*}\text{initial} = (7, 13) \qquad \qquad \quad \ \text{terminal} = (11, 19)\end{align*}$
  ::初始=( 7, 13) 终端=( 11, 19)

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  The vector needs to be translated to (0,0). Also, recall that magnitudes are always positive.
  ::矢量需要转换为(0,0),还要提醒注意,数值总是正值。

  $\begin{align*}(7, 13) + (-7, -13) = (0, 0)\qquad \qquad (11, 19) + (-7, -13) = (4, 6)\end{align*}$

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  horizontal $\begin{align*}= 4\end{align*}$ , vertical $\begin{align*}= 6\end{align*}$
  ::水平=4, 垂直=6

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  Example 4
  ::例4

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  Find the magnitude of the horizontal and vertical components of the following vector given the following coordinates of their initial and terminal points.
  ::根据下列初始点和终点点的坐标,查找下列矢量的横向和垂直组成部分的大小。

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  $\begin{align*}\text{initial} = (4.2, -6.8) \qquad \quad \ \text{terminal} = (-1.3, -9.4)\end{align*}$
  :4.2,-6.8)终端=(-1.3,-9.4)

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  The vector needs to be translated to (0,0). Also, recall that magnitudes are always positive.
  ::矢量需要转换为(0,0),还要提醒注意,数值总是正值。

  $\begin{align*}(4.2, -6.8) + (-4.2, 6.8) = (0, 0) \qquad \qquad (-1.3, -9.4) + (-4.2, 6.8) = (-5.5, -2.6)\end{align*}$

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  horizontal $\begin{align*}= 5.5\end{align*}$ , vertical $\begin{align*}= 2.6\end{align*}$
  ::水平=5.5, 垂直=2.6

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  Review
  ::回顾

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  In each question below, the initial and terminal coordinates for a vector are given. If the vector is translated so that it is in standard position (with the initial point at the origin), what are the new terminal coordinates?
  ::在下面的每个问题中,给出矢量的初始坐标和终端坐标。如果矢量被翻译为处于标准位置(初始点在源头),新的终端坐标是什么?

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  1. initial (2, 5) and terminal (7, -1)
     :2,5)和终点站(7,-1)
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  2. initial (4, 3) and terminal (3, -5)
     ::初始(4,3)和终端(3,5)
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  3. initial (8, 1) and terminal (-4, 7)
     ::初始(8,1)和终端(4,7)
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  4. initial (-2, 7) and terminal (3, 5)
     ::初始(-2,7)和终端(3,5)
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  5. initial (4, -3) and terminal (4, 3)
     ::初始(4,3)和终端(4,3)
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  6. initial (0, 2) and terminal (6, -4)
     ::初始(0,2)和终端(6,4)

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  Find the slope of each vector below with the given terminal coordinates. Assume the vector is in standard position.
  ::以给定的终端坐标查找下方每个矢量的斜度。 假设矢量处于标准位置 。

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  7. terminal (6, 7)
     ::终端终端(6,7)
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  8. terminal (3, 6)
     :3,6)终端(3,6)
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  9. terminal (-2, 4)
     :终端站-2,4)
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  10. terminal (5, 8)
      :5,8)终端(5,8)
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  11. terminal (1, 3)
      ::终端(1,3)

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  Find the magnitude of the horizontal and vertical components of each vector given the coordinates of their initial and terminal points.
  ::根据每个矢量初始点和终点点的坐标,查找每个矢量的水平和垂直组成部分的大小。

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  12. initial (1, 5) and terminal (1, -3)
      ::和终端(1, 5)和终端(1, 3)
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  13. initial (4, 5) and terminal (6, -5)
      ::初始(4,5)和终端(6,5)
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  14. initial (6, 1) and terminal (-4, 4)
      ::初始(6,1)和终端(4,4)
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  15. initial (-2, 3) and terminal (2, 5)
      ::初始(-2,3)和终端(2,5)

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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 。