Section: 两个矢量之间的点产品和角 | 7.4 两个矢量之间的点产品和角

Section outline

While two vectors cannot be strictly multiplied like numbers can, there are two different ways to find the product between two vectors. The between two vectors results in a new vector perpendicular to the other two vectors. You can study more about the cross product between two vectors when you take Linear Algebra. The second type of product is the dot product between two vectors which results in a regular number. Other names for the dot product include inner product and scalar product . This number represents how much of one vector goes in the direction of the other . In one sense, it indicates how much the two vectors agree with each other. This concept will focus on the dot product between two vectors.
::虽然两个矢量不能严格地像数字一样乘以两个矢量,但在两个矢量之间找到产品有两种不同的方式。两个矢量之间的两种矢量导致一个新的矢量与另外两个矢量垂直。在使用线性代数时,可以更多地研究两个矢量之间的交叉产物。第二种产品是两个矢量之间的点产物,得出一个正数。点产物的其他名称包括内产物和卡路里产物。这个数字表示一个矢量向另一个矢量的方向移动了多少。从某种意义上讲,它表明两个矢量之间的一致程度。这个概念将侧重于两个矢量之间的点产物。

What is the dot product between $\begin{aligned} < - 1, 1 > \end{aligned}$ and $\begin{aligned} < 4, 4 > \end{aligned}$ ? What does the result mean?
::#% 1, 1 > 和 < 4, 4> 之间的点产物是什么? 结果意味着什么 ?

Properties of the Dot Product
::点产品属性

The dot product is defined as:
::点产品的定义是:

\begin{aligned} u \cdot v =< u_{1}, u_{2} > \cdot < v_{1}, v_{2} >= u_{1} v_{1} + u_{2} v_{2} \end{aligned}

::uvu1,u22v1,v2u1v1+u2v2

This procedure states that you multiply the corresponding values and then sum the resulting products. It can work with vectors that are more than two dimensions in the same way.
::此程序规定, 您将相应的值乘以相应的值, 然后将生成的产品相加。它可以用相同的方式对两个以上维度以上的矢量起作用。

Before trying this procedure with specific numbers, look at the following pairs of vectors and relative estimates of their dot product.
::在用具体数字尝试这一程序之前,请先看看以下几对矢量及其点产品相对估计值。

Notice how vectors going in generally the same direction have a positive dot product. Think of two forces acting on a single object. A positive dot product implies that these forces are working together at least a little bit. Another way of saying this is the angle between the vectors is less than $\begin{aligned} 90^{\circ} \end{aligned}$ .
::注意通向同一方向的矢量通常有一个正点产物。想象一下两个力量在一个对象上行动。积极的点产物意味着这些力量至少合作了一点点。另一种说法是矢量之间的角小于 90\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

There are a many important properties related to the dot product. The two most important are 1) what happens when a vector has a dot product with itself and 2) what is the dot product of two vectors that are perpendicular to each other.
::圆点产品有许多重要的特性。最重要的两个特性是1) 当矢量本身有一个点产品时会发生什么,(2) 两个相互垂直的矢量的点产品是什么。

$\begin{aligned} v \cdot v = | v |^{2} \end{aligned}$
::{\fn方正黑体简体\fs18\b1\bord1\shad1\3cH2F2F2F2F2F2F2F2F}{\fn方正黑体简体\fs18\b1\bord1\shad1\3cH2F2F2F2F2F2

$\begin{aligned} v \end{aligned}$ and $\begin{aligned} u \end{aligned}$ are perpendicular if and only if $\begin{aligned} v \cdot u = 0 \end{aligned}$
::v 和 u 是直视的,如果且仅在v u=0的情况下。

The commutative property, $\begin{aligned} u \cdot v = v \cdot u \end{aligned}$ , holds for the dot product between two vectors. The following proof is for two dimensional vectors although it holds for any dimensional vectors.
::通量属性 uv=vu 持有两个矢量之间的点产物。以下为两个维矢量的证据,尽管它持有任何维矢量。

Start with the vectors in component form .
::从组件形式的矢量开始。

\begin{aligned} u & =< u_{1}, u_{2} > \\ v & =< v_{1}, v_{2} > \end{aligned}

::uu1,u2>vv1,v2>

Then apply the definition of dot product and rearrange the terms. The commutative property is already known for regular numbers so we can use that.
::然后应用点产品的定义, 并重新排列术语。通货产权已经以固定编号而为人所知, 这样我们就可以使用它。

\begin{aligned} u \cdot v & =< u_{1}, u_{2} > \cdot < v_{1}, v_{2} > \\ = u_{1} v_{1} + u_{2} v_{2} \\ = v_{1} u_{1} + v_{2} u_{2} \\ =< v_{1}, v_{2} > \cdot < u_{1}, u_{2} > \\ = v \cdot u \end{aligned}

::uvu1,u21,u2v1,v2u1+u2v2=v1u1+v2u21,v221,v22,u1,u2>=vu

The distributive property, $\begin{aligned} u \cdot (v + w) = u v + u w \end{aligned}$ , also holds under the dot product . The following proof will work with two dimensional vectors although the property does hold in general.
::分配属性 u(v+w) = uv+uw 也保留在点产品之下。以下证据将适用于两个维矢量,尽管该属性在一般情况下有效。

$\begin{aligned} u =< u_{1}, u_{2} >, v =< v_{1}, v_{2} >, w =< w_{1}, w_{2} > \end{aligned}$
::uu1,u2>,vv1,v2>,ww1,w2>

\begin{aligned} u \cdot (v + w) & = u \cdot (< v_{1}, v_{2} > + < w_{1}, w_{2} >) \\ = u \cdot < v_{1} + w_{1}, v_{2} + w_{2} > \\ =< u_{1}, u_{2} > \cdot < v_{1} + w_{1}, v_{2} + w_{2} > \\ = u_{1} (v_{1} + w_{1}) + u_{2} (v_{2} + w_{2}) \\ = u_{1} v_{1} + u_{1} w_{1} + u_{2} v_{2} + u_{2} w_{2} \\ = u_{1} v_{1} + u_{2} v_{2} + u_{1} w_{1} + u_{2} w_{2} \\ = u \cdot v + v \cdot w \end{aligned}

::u(v+w) =u( <v1,v2%w1,w2>) =uv1+w1,v2+w2_u1,u21,v2+w2+w1+w1+u2(v1+w1)+u2(v2+w2) =u1+u1+w1+u2+u2+w2+w2=u1+v1+u2+w2+w2=uv1+v2+w2+w2=uv2+v2+v2+wwww

The dot product can help you determine the angle between two vectors using the following formula. Notice that in the numerator the dot product is required because each term is a vector. In the denominator only regular multiplication is required because the magnitude of a vector is just a regular number indicating length.
::点产品可以帮助您使用以下公式确定两个矢量之间的角。注意, 在分子中需要点产品, 因为每个词都是矢量。在分母中只需要常规的乘法, 因为矢量的大小只是一个普通数字, 表示长度。

\begin{aligned} \cos θ = \frac{u \cdot v}{| u | | v |} \end{aligned}

::{\fn华文楷体\fs16\1cHE0E0E0}

Watch the portion of this video focusing on the dot product:
::观看这段片段的焦点是点产品:

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the dot product between the two vectors $\begin{aligned} < - 1, 1 > \end{aligned}$ and $\begin{aligned} < 4, 4 > \end{aligned}$ . It can be computed as:
::早些时候, 您被要求在两个矢量 1, 1 > 和 < 4, 4 > 之间找到点产物。可以计算为 :

$\begin{aligned} (- 1) (4) + 1 (4) = - 4 + 4 = 0 \end{aligned}$

The result of zero makes sense because these two vectors are perpendicular to each other.
::零的结果是有道理的,因为这两个矢量是相互垂直的。

Example 2
::例2

Find the dot product between the following vectors: $\begin{aligned} < 3, 1 > \cdot < 5, - 4 > \end{aligned}$
::在以下矢量中查找点产品: < 3,15,-4>

$\begin{aligned} u =< 3, 5 > \end{aligned}$
::3,5>

Example 3
::例3

Prove the angle between two vectors formula:
::证明两个矢量公式之间的角 :

$\begin{aligned} \cos θ = \frac{u \cdot v}{| u | | v |} \end{aligned}$
::{\fn华文楷体\fs16\1cHE0E0E0}

Start with the law of cosines.
::先从鱼定律开始

\begin{aligned} | u - v |^{2} & = | v |^{2} + | u |^{2} - 2 | v | | u | \cos θ \\ (u - v) \cdot (u - v) & = \\ u \cdot u - 2 u \cdot v + v \cdot v & = \\ | u |^{2} - 2 u \cdot v + | v |^{2} & = \\ - 2 u \cdot v & = - 2 | v | | u | \cos θ \\ \frac{u \cdot v}{| u | | v |} & = \cos θ \end{aligned}

::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ( ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Example 4
::例4

Find the dot product between the following vectors.
::在以下矢量之间找到点产品。

$\begin{aligned} (4 i - 2 j) \cdot (3 i - 8 j) \end{aligned}$
:4i-2j) (3i-8j)

The standard unit vectors can be written as component vectors.
::标准单位矢量可以作为组成矢量书写。

$\begin{aligned} < 4, - 2 > \cdot < 3, - 8 >= 12 + (- 2) (- 8) = 12 + 16 = 28 \end{aligned}$

Example 5
::例5

What is the angle between $\begin{aligned} u =< 3, 5 > \end{aligned}$ and $\begin{aligned} v =< 2, 8 > \end{aligned}$ ?
::u3-5> 与v2,8> 之间的角是什么?

Use the angle between two vectors formula.
::使用两个矢量公式之间的角。

$\begin{aligned} u =< 3, 5 > \end{aligned}$ and $\begin{aligned} v =< 2, 8 > \end{aligned}$
::3-5>和v2,8>

\begin{aligned} \frac{u \cdot v}{| u | | v |} & = \cos θ \\ \frac{< 3, 5 > \cdot < 2, 8 >}{\sqrt{34} \cdot \sqrt{68}} & = \cos θ \\ \frac{6 + 40}{\sqrt{34} \cdot \sqrt{68}} & = \cos θ \\ \cos^{- 1} (\frac{46}{\sqrt{34} \cdot \sqrt{68}}) & = θ \\ {16.93}^{\circ} & \approx θ \end{aligned}

::2,8>34 68= 6+4034 68= 1 (46346816.93

Summary

The dot product, also known as inner product or scalar product, is a way to find the product between two vectors, resulting in a regular number.

$\begin{aligned} u \cdot v =< u_{1}, u_{2} > \cdot < v_{1}, v_{2} >= u_{1} v_{1} + u_{2} v_{2} \end{aligned}$

::圆点产品,也称为内产产品或卡路里产品,是两种矢量之间找到该产品的一种方法,结果得出一个常规数字。

Some properties of dot products include
- $\begin{aligned} v \cdot v = | v |^{2} \end{aligned}$
  ::{\fn方正黑体简体\fs18\b1\bord1\shad1\3cH2F2F2F2F2F2F2F2F}{\fn方正黑体简体\fs18\b1\bord1\shad1\3cH2F2F2F2F2F2
- $\begin{aligned} v \end{aligned}$ and $\begin{aligned} u \end{aligned}$ are perpendicular if and only if $\begin{aligned} v \cdot u = 0. \end{aligned}$
  ::v 和 u 具有直视性,如果且仅在v u=0时才具有直视性。
- $\begin{aligned} u \cdot v = v \cdot u \end{aligned}$
  ::uv=vu
- $\begin{aligned} v \cdot (v + w) = u v + v w \end{aligned}$
  ::v(v+w) = uv+vw
::圆点产品的某些属性包括 vvv2 v 和 u 只有在 vu=0. uv=vu vv=uv+vw 时才具有垂直特性

The dot product can be used to determine the angle between two vectors using the formula: $\begin{aligned} \cos θ = \frac{u \cdot v}{| u | | v |} . \end{aligned}$
::点产品可用于使用公式确定两个矢量之间的角: cosuvuv。

Review
::回顾

Find the dot product for each of the following pairs of vectors.
::为以下每种矢量寻找点产品。

1. $\begin{aligned} < 2, 6 > \cdot < - 3, 5 > \end{aligned}$

2. $\begin{aligned} < 5, - 1 > \cdot < 4, 4 > \end{aligned}$

3. $\begin{aligned} < - 3, - 4 > \cdot < 2, 2 > \end{aligned}$

4. $\begin{aligned} < 3, 1 > \cdot < 6, 3 > \end{aligned}$

5. $\begin{aligned} < - 1, 4 > \cdot < 2, 9 > \end{aligned}$

Find the angle between each pair of vectors below.
::查找下方每对矢量之间的角。

6. $\begin{aligned} < 2, 6 > \cdot < - 3, 5 > \end{aligned}$

7. $\begin{aligned} < 5, - 1 > \cdot < 4, 4 > \end{aligned}$

8. $\begin{aligned} < - 3, - 4 > \cdot < 2, 2 > \end{aligned}$

9. $\begin{aligned} < 3, 1 > \cdot < 6, 3 > \end{aligned}$

10. $\begin{aligned} < - 1, 4 > \cdot < 2, 9 > \end{aligned}$

11. What is $\begin{aligned} v \cdot v \end{aligned}$ ?
::11. 什么是Vv?

12. How can you use the dot product to find the magnitude of a vector?
::12. 您如何使用点产品找到矢量的大小?

13. What is $\begin{aligned} 0 \cdot v \end{aligned}$ ?
::13. 什么是0v?

14. Show that $\begin{aligned} (c u) \cdot v = u \cdot (c v) \end{aligned}$ where $\begin{aligned} c \end{aligned}$ is a constant.
::14. 显示 c 是常数的 (cu)v=u(cv) 。

15. Show that $\begin{aligned} < 2, 3 > \end{aligned}$ is perpendicular to $\begin{aligned} < 1.5, - 1 > \end{aligned}$ .
::15. 显示 <2,3>与 <1.5,-1> 垂直。

Review (Answers)
::回顾(答复)

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。

Section outline

Properties of the Dot Product ::点产品属性

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)