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  • 将矢量溶解成组件

    Resolving Vectors into Axial Components
    ::将矢量溶解为轴构件

    We know that when two are in the same dimension, they can be added arithmetically.  Suppose we have two vectors that are on a north-south, east-west grid, as shown below.  One of the methods we can use to add these vectors is to resolve each one into a pair of vectors that lay on the north-south and east-west axes. 
    ::我们知道,当两个维度相同时,它们可以用算法来添加。假设我们有两个矢量,它们位于南北、东西方网格上,如下文所示。我们可以将这些矢量添加的方法之一是将每种矢量溶解为位于南北和东西方轴上的一对矢量。

    Two vectors are shown: one at 30° north of east (65 N) and another at 60° north of west (35 N).

    The two vectors we are to add is a force of 65 N at 30° north of east and a force of 35 N at 60° north of west.
    ::我们要补充的两个矢量是东经30度以北65牛顿,西经60度以北35牛顿。

    Vectors showing force components at specified angles, labeled with magnitudes and orientations.

    We can resolve each of the vectors into two components on the axes lines. Each vector is resolved into a component on the north-south axis and a component on the east-west axis.
    ::我们可以将每种矢量分为轴线的两个组成部分,将每种矢量分为南北轴的一个组成部分和东西轴的一个组成部分。

    Using trigonometry, we can resolve (break down) each of these vectors into a pair of vectors that lay on the axial lines (shown in red above).
    ::使用三角测量法,我们可以解析(分解)这些矢量中的每一种,将其分为一对位于轴线上的矢量(如上面红色所示)。

    The east-west component of the first vector is (65 N)(cos 30° ) = (65 N)(0.866) = 56.3 N east
    ::第一个矢量的东-西部分为(65牛顿(30°)(65牛顿)(0.866)=56.3牛顿(东经)(65牛顿)(30°)=(65牛顿)(0.866)=56.3牛顿(东经)。

    The north-south component of the first vector is (65 N)(sin 30°) = (65 N)(0.500) = 32.5 N north
    ::第一个矢量的北向-南部分为(65牛顿)(30°)(65牛顿)(0.500)=32.5牛顿(北纬)=(65牛顿)(30°)(65牛顿)(0.500)=32.5牛顿)。

    The east-west component of the 2 nd vector is (35 N)(cos 60°) = (35 N)(0.500) = 17.5 N west
    ::第二矢量的东-西部分为(35牛顿)(60°)=(35牛顿)(0.500)=17.5牛顿西部。

    The north-south component of the 2 nd vector is (35 N)(sin 60°) = (35 N)(0.866) = 30.3 N north
    ::第二矢量的北向-南部分为(35牛顿)(60°)(35牛顿)(0.866)=30.3牛顿北部。

    E xplore how to resolve vectors  into axial components by hanging portraits of legendary physicists in the simulation below. The angle of each string determines how its tension force is distributed in both the horizontal and vertical dimensions. Under what conditions will the strings holding the portrait snap?
    ::探索如何在下面的模拟中挂上传说物理学家的肖像, 将矢量解为轴元件。 每个字符串的角度决定着它的张力如何在水平和垂直维度上分布。 在什么条件下, 持有肖像的字符串将会在什么条件下松动 ?

    Summary
    ::摘要

    • Vectors can be resolved into component vectors that lie on the axes lines.
      ::矢量可以溶解成在轴线上的成份矢量。

    Review
    ::回顾

    1. A force of 150. N is exerted 22° north of east.  Find the northward and eastward components of this force.
      ::N部队在东面以北22度处投放,向北和向东部署。
    2. An automobile travels a displacement of 75 km 45° north of east.  How far east does it travel and how far north does it travel?
      ::一辆汽车在东面以北45度以北75公里处行驶,东面行驶多远,北面行驶多远?

    Explore More
    ::探索更多

    Use this resource to answer the questions that follow.
    ::使用此资源回答下面的问题 。

    1. What does SohCahToa mean? 
      ::苏加托是什么意思?
    2. Why is SohCahToa relevant to resolving a vector into components?
      ::为什么SohCahToa与将矢量溶解成组成部分有关?
    3. Why is the sum of the components larger than the resultant vector?
      ::为什么各组成部分的总和大于由此产生的矢量?