Section: 摘要:矢量 | 9.9 摘要:矢量

Section outline

Physicists, video game designers, meteorologists, and air traffic controllers all use vectors. Vector quantities have a direction and a magnitude. In this chapter, you learned that you can add and subtract vectors just as you can numbers. You saw that vectors also have other algebraic properties and essentially create an entirely new algebraic structure. There are two ways the product of vectors can be taken, and you studied one way called the dot product. Lastly, you learned to calculate how much of one vector goes in the direction of another vector.

Chapter Summary
::章次摘要

In this chapter we learned:
::在本章中,我们学到:

A vector is a set of instructions indicating direction and magnitude.
::矢量是一组指示,指示方向和数量。

A vector is determined by two points: the initial point, called the tail, and its terminal point, called the head.
::矢量由两个点确定:初始点,称为尾端,及其终点,称为头部。

A force diagram is a collection of vectors where each represents a force, like gravity or wind, acting on an object.
::力图是矢量的集合,其中每个矢量代表一种力量,如重力或风力,在物体上活动。

Magnitude refers to the length of the vector and is associated with the strength of the force or the speed of the object.
::磁度指矢量的长度,并与物体的强度或速度相关联。

Bearing is measured with 0 ^∘ as due North, 90 ^∘ as East, 180 ^∘ as South, and 270 ^∘ as West.
::以0为北,90为东,180为南,270为西测量。

The magnitude of the position vector is given by

$\begin{aligned} | \vec{P} | = \sqrt{P_{x}^{2} + P_{y}^{2}} \end{aligned}$

::PPx2+Py2

::位置矢量的大小由 PPPx2+Py2 给出

A resultant vector is the vector that is produced when two or more vectors are summed or subtracted. It is also what is produced when a single vector is scaled by a constant.
::由此产生的矢量是指在对两个或两个以上矢量进行总和或减去时产生的矢量,也是指单个矢量用常数缩放时产生的矢量。

To scale a vector, multiply the vector's magnitude and direction by the common scale factor.
::要缩放矢量,将矢量的大小和方向乘以共同的尺度系数。

A unit vector is a vector of magnitude 1.
::单位矢量是星级 1 的矢量。

Component form means in the form $\begin{aligned} < x, y > \end{aligned}$ .
::元件窗体是指窗体 <x,y> 中的窗体。

The standard unit vectors are $\begin{aligned} \vec{i}, \end{aligned}$ which is the vector $\begin{aligned} < 1, 0 >, \end{aligned}$ and $\begin{aligned} \vec{j}, \end{aligned}$ which is the vector $\begin{aligned} < 0, 1 > \end{aligned}$ .
::标准单位矢量是i,即矢量 < 1,0 > 和j,即矢量 < 0,1 > 。

The dot product is also known as inner product and scalar product. It is one of two kinds of products taken between vectors. It produces a number that can be interpreted to tell how much one vector goes in the direction of the other.
::圆点产品也称为内产物和卡路里产品,是介于矢量之间的两种产品之一,可解释为能说明一个矢量向另一个矢量的方向移动多少。

The dot product formula is $\begin{aligned} u \cdot v =< u_{1}, u_{2} > \cdot < v_{1}, v_{2} >= u_{1} v_{1} + u_{2} v_{2} . \end{aligned}$
::圆点产品公式为 uvu1,u2v1,v2u1+u2v2。

The vector projection of a vector onto a given direction has a magnitude equal to the scalar projection.
::向某一方向向矢量投射的矢量的大小与标量投射的数值相等。

A scalar projection is given by the dot product of a vector, with a unit vector for that direction.
::矢量的点产物给出了标量预测,该方向的单位矢量为单位矢量。

Scalar projection is the length of the vector projection.
::Scalar 预测是矢量预测的长度。

The vector projection formula is $\begin{aligned} p r o j_{u} v = (\frac{u \cdot v}{| u |}) \frac{u}{| u |} \end{aligned}$ .
::矢量投影公式为 projuv= (uvu) uu。

The vector form of the equation of a line is $\begin{aligned} \vec{r} = \vec{r_{0}} + t \vec{v} =< x_{0}, y_{0} > + t < a, b > . \end{aligned}$
::线条方程式的矢量形式为 r0tvx0,y0t<a,b>。

Review
::回顾

Try the following cumulative review problems to practice the concepts studied in this chapter:
::尝试下列累积审查问题,以实践本章所研究的概念:

Section outline

Chapter Summary ::章次摘要

Review ::回顾

Chapter Summary
::章次摘要

Review
::回顾