Course: 5.19 向量和组件 | The Way To Learn

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section 向量和组件

Collapse Expand
向量和组件
While working in your math class at school, the instructor passes everyone a map of Your town. She asks you to find your house and place a red dot on it, and then find the school and place a blue dot there. Your map looks like this:
::老师在学校的数学课上工作时,给每个人一张你镇上的地图。她要你找到你的房子,在上面放一个红色的点,然后找到学校,在那里放一个蓝色的点。你的地图是这样的:

She then asks you to break down the trip to your school in terms of component vectors and unit vectors. Are you able to do this?
::然后她要求你分解去学校的行程,从组件矢量和单位矢量来看。你能做到这一点吗?

Unit Vectors and Components
::向量和组件

A unit vector is a vector that has a magnitude of one unit and can have any direction. Traditionally $\begin{aligned} \hat{i} \end{aligned}$ (read “ $\begin{aligned} i \end{aligned}$ hat”) is the unit vector in the $\begin{aligned} x \end{aligned}$ direction and $\begin{aligned} \hat{j} \end{aligned}$ (read “ $\begin{aligned} j \end{aligned}$ hat”) is the unit vector in the $\begin{aligned} y \end{aligned}$ direction. $\begin{aligned} | \hat{i} | = 1 \end{aligned}$ and $\begin{aligned} | \hat{j} | = 1 \end{aligned}$ . Unit vectors on perpendicular axes can be used to express all vectors in that plane. Vectors are used to express position and motion in three dimensions with $\begin{aligned} \hat{k} \end{aligned}$ (“ $\begin{aligned} k \end{aligned}$ hat”) as the unit vector in the $\begin{aligned} z \end{aligned}$ direction.
::单位矢量是指具有一个单位大小并且可以有任何方向的矢量。传统上, i (读“i hat”) 是指 x 方向的单位矢量, j (读“j hat”) 是指 y 方向的单位矢量。 1 和 j1 。垂直轴上的单位矢量可用于表达该平面上的所有矢量。矢量用于表达三个维的方位和运动, k (k hat) 是 z 方向的单位矢量。

We are not studying 3D space in this course. The unit vector notation may seem burdensome but one must distinguish between a vector and the components of that vector in the direction of the $\begin{aligned} x - \end{aligned}$ or $\begin{aligned} y - \end{aligned}$ axis. The unit vectors carry the meaning for the direction of the vector in each of the coordinate directions. The number in front of the unit vector shows its magnitude or length. Unit vectors are convenient if one wishes to express a 2D or 3D vector as a sum of two or three orthogonal components, such as $\begin{aligned} x - \end{aligned}$ and $\begin{aligned} y - \end{aligned}$ axes, or the $\begin{aligned} z - \end{aligned}$ axis. (Orthogonal components are those that intersect at right angles.)
::我们没有研究此课程中的 3D 空间。单位矢量标记可能看起来很麻烦, 但必须区分矢量和该矢量在 x - 或 y - 轴方向的成分。单位矢量在每个坐标方向都有矢量方向的含义。单位矢量前面的数值显示其大小或长度。单位矢量如果想将 2D 或 3D 矢量表示为两个或三个正方位的成分之和, 如 x - 和 y - 轴, 或 z - 轴。 (正弦矢量是那些在右角交叉的成分。)

Component vectors of a given vector are two or more vectors whose sum is the given vector. The sum is viewed as equivalent to the original vector. Since component vectors can have any direction, it is useful to have them perpendicular to one another. Commonly one chooses the $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ axis as the basis for the unit vectors. Component vectors do not have to be orthogonal.
::给定矢量的成份矢量是两个或两个以上的矢量,其总和是给定矢量。该总和被视为与原矢量相等。由于成份矢量可以有任何方向,因此让它们相互垂直是有用的。通常选择 x 和 y 轴作为单位矢量的基础。成份矢量不必是正向的。

A vector from the origin (0, 0) to the point (8, 0) is written as $\begin{aligned} 8 \hat{i} \end{aligned}$ . A vector from the origin to the point (0, 6) is written as $\begin{aligned} 6 \hat{j} \end{aligned}$ .
::从源(0,0)到点(8,0)的矢量写成 8i。从源(0,6)到点(0,6)的矢量写成 6j。

The reason for having the component vectors perpendicular to one another is that this condition allows us to use the Pythagorean Theorem and trigonometric ratios to find the magnitude and direction of the components. One can solve vector problems without use of unit vectors if specific information about orientation or direction in space such as N, E, S or W is part of the problem.
::使组成矢量相互垂直的原因是,这一条件允许我们使用Pythagorean理论和三角比来找到组成物的大小和方向;如果关于空间方向或方向的具体信息,如N、E、S或W是问题的一部分,那么不用单位矢量就可以解决矢量问题。

Let's take a look at a few example problems.
::让我们来看一下几个例子问题。

1. What are the component vectors of the vector shown here?
::1. 这里显示矢量的构成矢量是什么?

Since the length of the vector is 5, and the angle the vector makes with the $\begin{aligned} x \end{aligned}$ axis is $\begin{aligned} {53.13}^{\circ} \end{aligned}$ , the "x" component of the vector is:
::由于矢量的长度为5,而矢量与x轴的角为53.13,矢量的“x”成分为:

$\begin{aligned} | V_{x} | = | \vec{V} | \cos {53.13}^{\circ} \\ | V_{x} | = (5) (.6) = 3 \end{aligned}$

::Vx*V*V*Cos*53.13*Vx*(5)(6)=3

And the "y" component is:
::"Y"组件是:

$\begin{aligned} | V_{y} | = | \vec{V} | \sin {53.13}^{\circ} \\ | V_{y} | = (5) (.8) = 4 \end{aligned}$

::维耶维什金53.13 维耶维(5)(8)=4

And we have the familiar 3, 4, 5 triangle, where the vector is the hypotenuse.
::我们有熟悉的3,4,5三角形, 矢量就是低温。

2. Why are unit vectors required when dealing with vector addition?
::2. 为什么在处理添加病媒时需要单位矢量?

Unit vectors are required because it is necessary to have like quantities for addition. If there are two numbers, they can be added. If there are two vectors, they can be added. But if you have a number and a vector, they can't be added. Having unit vectors along with a magnitude makes a quantity a vector.
::需要单位矢量是因为需要同样数量来添加。如果有两个数字, 可以添加它们。如果有两个矢量, 可以添加它们。但是, 如果您有数字和一个矢量, 则不能添加它们。如果有单位矢量加上一个数量, 则一个数量是一个矢量。

3. What are the unit vectors and the lengths of the component vectors when
::3. 当组件矢量的单位矢量和长度

$\begin{aligned} \vec{V} = 7 \hat{i} + 9 \hat{j} \end{aligned}$
::

The unit vectors in this case are $\begin{aligned} \hat{i} \end{aligned}$ and $\begin{aligned} \hat{j} \end{aligned}$ . In some courses and books you might see the notation for unit vectors written instead as $\begin{aligned} \hat{x} \end{aligned}$ and $\begin{aligned} \hat{y} \end{aligned}$ .
::单位矢量在此情况下是 i 和 j 。在有些课程和书籍中, 您可能会看到单位矢量的标记, 代之以 x 和 y 。

The length of the in the $\begin{aligned} \hat{i} \end{aligned}$ direction is 7, and the component vector in the $\begin{aligned} \hat{j} \end{aligned}$ direction is 9.
::i 方向的长度为 7, j方向的组件矢量为 9。

Examples
::实例

Example 1
::例1

Earlier, you were asked to break down the trip to your school in terms of component vectors and unit vectors.
::早些时候,你被要求分解去你学校的行程从组件矢量和单位矢量。

In this section, you learned that breaking a vector down into its components involves adding the portion of the vector along the "y" axis to the portion of the vector along the "x" axis. To accomplish this in the case of the map, you only need to write down the length the vector has in the "x" direction (along with an "x" unit vector) and then add to it the length the vector has in the "y" direction (along with a "y" unit vector). Your map should look like this:
::在本节中,您了解到,将矢量向下折入其组件意味着将矢量部分沿着“y”轴添加到“x”轴的矢量部分。要做到这一点,在映射的情况下,只需将矢量在“x”方向(连同“x”单位矢量)上的长度写下来,然后将矢量在“y”方向(连同“y”单位矢量)的长度添加到其中。您的映射应该像这样看 :

Example 2
::例2

An inclined ramp is 12 feet long and forms an angle of $\begin{aligned} {28.2}^{\circ} \end{aligned}$ with the ground. Find the horizontal and vertical components of the ramp.
::斜坡长12英尺,与地面形成28.2角。找到斜坡的水平和垂直部分。

$\begin{aligned} y = 12 \sin {28.2}^{\circ} = 5.7, x = 12 \cos {28.2}^{\circ} = 10.6 \end{aligned}$
::y=12sin 28.25.7,x=12cos 28.210.6

Example 3
::例3

A wind vector has a magnitude of 25 miles per hour with an angle of $\begin{aligned} 20^{\circ} \end{aligned}$ with respect to the east. Determine how much the wind is blowing to the north and how much it is blowing to the east.
::风向矢量为每小时25英里,以东方角度为20°C。确定风向向北吹多少风,向东吹多少风。

Since the vector has an angle of $\begin{aligned} 20^{\circ} \end{aligned}$ with respect to the horizontal, the component to the east is $\begin{aligned} 25 \cos 20^{\circ} = 23.49 \end{aligned}$ miles per hour. In the same way, the component to the north is $\begin{aligned} 25 \sin 20^{\circ} = 8.55 \end{aligned}$ miles per hour.
::由于矢量在水平上的角度为20°C, 东面的成分为每小时25°C 20°C 23.49英里, 北面的成分为每小时25°C 20°C 8.55英里。

Example 4
::例4

A vector $\begin{aligned} | \vec{V} | \end{aligned}$ has a magnitude of 25 inches, and is at an angle of $\begin{aligned} 80^{\circ} \end{aligned}$ with respect to the positive "x" axis. Write the vector in component and unit vector notation.
::矢量 V 的大小为 25 英寸,对于正“x”轴,其角度为 80 英寸。将矢量写在组件和单位矢量符号中。

The "x" component is $\begin{aligned} 25 \cos 80^{\circ} = 4.34 \end{aligned}$ . The "y" component is $\begin{aligned} 25 \sin 80^{\circ} = 24.62 \end{aligned}$ . Therefore, the vector can be written as $\begin{aligned} | \vec{V} | = 4.34 \hat{i} + 24.62 \hat{j} \end{aligned}$ .
::“x”组件为 25cos804.34。“y”组件为 25sin8024.62。因此,矢量可以写为 {V}4.34i}24.62j}。

Review
::回顾

Describe how to find the vertical and horizontal components of a vector when given the magnitude and direction of the vector.
::根据矢量的大小和方向,描述如何找到矢量的垂直和水平组成部分。

$\begin{aligned} \vec{a} \end{aligned}$ has a magnitude of 6 and a direction of $\begin{aligned} 100^{\circ} \end{aligned}$ . Find the components of the vector.
::a 的大小为 6, 方向为 100 。找到矢量的组件。

$\begin{aligned} \vec{b} \end{aligned}$ has a magnitude of 3 and a direction of $\begin{aligned} 60^{\circ} \end{aligned}$ . Find the components of the vector.
::b 的大小为 3, 方向为 60 。找到矢量的组件。

$\begin{aligned} \vec{c} \end{aligned}$ has a magnitude of 2 and a direction of $\begin{aligned} 84^{\circ} \end{aligned}$ . Find the components of the vector.
::c 的大小为 2, 方向为 84 。找到矢量的成分。

$\begin{aligned} \vec{d} \end{aligned}$ has a magnitude of 5 and a direction of $\begin{aligned} 32^{\circ} \end{aligned}$ . Find the components of the vector.
::d 的大小为 5, 方向为 32。找到矢量的成分。

$\begin{aligned} \vec{e} \end{aligned}$ has a magnitude of 2 and a direction of $\begin{aligned} 45^{\circ} \end{aligned}$ . Find the components of the vector.
::e 的大小为 2, 方向为 45。找到矢量的组件。

$\begin{aligned} \vec{f} \end{aligned}$ has a magnitude of 7 and a direction of $\begin{aligned} 70^{\circ} \end{aligned}$ . Find the components of the vector.
::f 的大小为 7, 方向为 70 。找到矢量的组件。

A plane is flying on a bearing of $\begin{aligned} 50^{\circ} \end{aligned}$ at 450 mph. Find the component form of the velocity of the plane. What does the component form tell you?
::一架飞机在450 mph. 找到飞机速度的部件形式。部件形式告诉你什么?

A baseball is thrown at a $\begin{aligned} 20^{\circ} \end{aligned}$ angle with the horizontal with an initial speed of 30 mph. Find the component form of the initial velocity.
::棒球被投向20°Z角,水平水平,初始速度为30mph。查找初始速度的组件形式。

A plane is flying on a bearing of $\begin{aligned} 300^{\circ} \end{aligned}$ at 500 mph. Find the component form of the velocity of the plane.
::一架飞机在500米时方位300°Z的方位飞行。找到飞机速度的部件形式。

A plane is flying on a bearing of $\begin{aligned} 150^{\circ} \end{aligned}$ at 470 mph. At the same time, there is a wind blowing at a bearing of $\begin{aligned} 200^{\circ} \end{aligned}$ at 60 mph. What is the component form of the velocity of the plane?
::一架飞机在470米时方位150°C的空中飞行。同时,在60米时方位200°C的空中有风吹。飞机速度的构成形式是什么?

Using the information from the previous problem, find the actual ground speed of the plane.
::利用上一个问题的信息,找到飞机的实际地面速度。

Wind is blowing at a magnitude of 50 mph with an angle of $\begin{aligned} 25^{\circ} \end{aligned}$ with respect to the east. What is the velocity of the wind blowing to the north? What is the velocity of the wind blowing to the east?
::东风以25°C的角吹向东面,风吹向北面的速度是多少? 风吹向东面的速度是多少? 风吹向东面的速度是多少?

Find a unit vector in the direction of $\begin{aligned} \vec{a} \end{aligned}$ , a vector in standard position with terminal point (-4, 3).
::向 a__ 方向查找一个单位矢量,一个具有终点标准位置的矢量(-4, 3) ,一个具有标准位置的矢量(-4, 3) 。

Find a unit vector in the direction of $\begin{aligned} \vec{b} \end{aligned}$ , a vector in standard position with terminal point (5, 1).
::在b方向上找到一个单位矢量,一个具有终点标准位置的矢量(5,1),具有终点点(5,1)。

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

General

向量和组件

Unit Vectors and Components ::向量和组件

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Unit Vectors and Components
::向量和组件

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)