Unit vector notation | Vectors and spaces | Linear Algebra | Khan Academy

Summary

这段视频讲解了如何使用单位向量i和j来表示二维向量。首先介绍了二维向量可以表示为一对有序数对(2,3)的形式,然后引入了使用单位向量和标量乘法的记法方式,即向量v可以表示为2i+3j的形式。接着通过实例演示了如何使用单位向量记法对两个向量进行加法运算。最后将向量用单位向量和标量乘法表示的结果与列向量表示进行了等同性的说明,阐明了两种记法之间的关系。总的来说,这一过程很好地解释了单位向量在表示和操作向量时的应用。

Takeaways

• 😀 向量可以用箭头来表示,箭头的长度代表向量的大小,方向代表向量的方向。
• 🤔 一个二维向量可以用有序对(x, y)来表示,其中x代表水平方向的分量,y代表垂直方向的分量。
• ✏️ 引入了单位向量 i 和 j 的概念,i 代表水平方向的单位向量,j 代表垂直方向的单位向量。
• ➕ 任何二维向量都可以表示为 i 和 j 的线性组合,即 v = xi + yj,其中 x 和 y 是相应的系数。
• 🔢 例如,向量 v = (2, 3) 可以写作 2i + 3j。
• ⚡ 用单位向量表示法可以方便地进行向量加法和数乘运算。
• ➕ 向量加法就是将对应分量相加:v + w = (v_x + w_x)i + (v_y + w_y)j。
• ✖️ 数乘就是将向量的每个分量乘以相同的数:kv = (kv_x)i + (kv_y)j。
• 🌐 单位向量表示法在高维情况下也适用,比如三维空间引入 k 作为第三个单位向量。
• 💡 这种表示方法有助于理解向量的本质,并为后续的向量运算奠定基础。

Q & A

1. 什么是向量的视觉表示?

向量可以用一个箭头来表示,箭头的长度表示向量的大小,箭头的方向表示向量的方向。

2. 如何用数学方式表示一个二维向量?

可以用一个有序对或二元组来表示,第一个数值表示水平方向的分量,第二个数值表示垂直方向的分量。例如向量v可以表示为(2,3)。

3. 什么是单位向量,如何表示?

单位向量是指在某个特定方向上长度为1的向量。在二维空间中,有两个单位向量i和j,分别对应水平和垂直方向。用一个小帽符号来表示,如i表示水平方向上的单位向量,j表示垂直方向上的单位向量。

4. 如何用单位向量表示一个向量?

任何二维向量都可以表示为单位向量i和j的缩放和。例如,向量v可以表示为2i + 3j,其中2是水平方向上的缩放系数,3是垂直方向上的缩放系数。

5. 如何用单位向量表示法进行向量加法运算?

向量加法可以通过将各个分量相加来实现。例如,v + b = (2 + (-1))i + (3 + 4)j = i + 7j。

6. 单位向量表示法和列向量表示之间有什么关系?

这两种表示法是等价的,可以相互转换。例如v可以写作(2, 3)的列向量形式,也可以写作2i + 3j的单位向量形式。

7. 单位向量表示法在三维空间中会有什么变化?

在三维空间中,除了i和j表示水平和垂直方向外,还需要引入第三个单位向量k表示第三个维度的方向。

8. 为什么需要引入单位向量的概念?

引入单位向量的概念是为了方便对向量进行缩放和相加等运算。它提供了一种标准的基础,使得向量运算更加统一和简洁。

9. 如何用单位向量表示法表示负方向的向量?

可以在单位向量前乘以负数来表示相反方向。例如,-2i表示水平方向上长度为2,方向相反的向量。

10. 单位向量表示法和传统的(x, y)坐标表示有何区别?

单位向量表示法更加抽象和一般化,不必局限于特定的坐标系。它提供了一种独立于坐标系的方式来表示和操作向量。(x, y)坐标表示隐含了特定的坐标系选择。

Outlines

00:00

🔢 向量的图形与数学表示

这一部分介绍了向量的基本概念，解释了如何通过箭头图形表示向量，其中箭头的长度代表向量的大小，方向代表向量的方向。进一步，展示了如何数学地表示向量，即通过确定向量头部在水平和垂直方向上相对于尾部的距离。例如，一个向右2个单位、向上3个单位的向量，可以表示为有序对(2, 3)。接着引入了单位向量的概念，定义了在二维空间中水平方向的单位向量i和垂直方向的单位向量j，并展示了如何用这些单位向量以及它们的倍数来表示任何二维向量。通过对单位向量的缩放和相加，可以构建出任意向量，例如将2倍的i向量和3倍的j向量相加来表示向量v。

05:00

📏 向量运算与单位向量表示法

这部分进一步探讨了使用单位向量表示法进行向量运算的方法。首先，介绍了如何将三维空间中的向量表示为单位向量i、j以及新引入的单位向量k的线性组合。然后，通过一个具体的例子，演示了如何使用单位向量表示法来进行向量加法。示例中定义了两个向量v和b，并询问向量v加上向量b的结果。通过直接将相应的单位向量成分相加，演示了向量加法的过程，并将结果以单位向量表示法以及列向量形式展现出来。这展示了单位向量表示法在简化向量运算中的效用，并且证明了不同表示法之间的一致性和转换的自然性。

Keywords

💡向量

向量是一个具有大小和方向的量。在视频中,向量通过箭头的长度表示大小,箭头的方向表示方向。例如,视频中讲到"箭头的长度就是向量的大小,方向就是向量的方向"。向量在视频中被用来表示平面或空间中的位移或位置。

💡单位向量

单位向量是大小为1的特殊向量。在视频中引入了i和j两个单位向量,分别代表沿x轴和y轴的单位向量。单位向量用来作为基础向量,可以通过缩放和相加来构造任意向量。视频举例说明了如何用2i+3j来表示向量(2,3)。

💡向量表示

视频介绍了几种向量的表示方式:箭头、有序数对(2,3)、列向量(2,3)^T以及单位向量的线性组合2i+3j。不同的表示方式都能唯一对应同一个向量,只是从不同角度描述了向量的信息。视频旨在说明单位向量组合的好处。

💡向量加法

向量加法即分量对应相加。视频以向量v=(2,3)和b=(-1,4)为例,演示了如何通过相加分量得到v+b的结果,并且指出这与单位向量表示下2i+3j+(-i+4j)的做法是等价的。向量加法体现了几何和代数角度的统一。

💡标量缩放

标量缩放指的是将一个向量乘以一个实数的操作。视频中通过2i和3j来表示向量(2,3),其中2和3就是缩放系数。标量缩放改变了向量的大小而不改变方向,是构造新向量的基本方式之一。

💡分量

向量的分量指的是向量在坐标系的每个坐标轴上的投影长度。在二维平面,一个向量通常由x和y两个分量确定。视频多次提到向量"在水平方向上的长度"和"在垂直方向上的长度",就是在描述向量的分量。分量提供了从几何到代数的过渡。

💡坐标系

坐标系为研究向量提供了背景框架。视频涉及的都是二维平面直角坐标系,其中包含互相垂直的x轴和y轴。单位向量i和j就是沿着这两个坐标轴的方向。而任意向量都可以用坐标对或单位向量线性组合在这个坐标系中表示。

💡基向量

基向量是指能够生成整个向量空间的一组基本向量的集合。在二维情况,i和j就构成了一组基向量。视频旨在阐述如何利用这两个基本单位向量,通过缩放和相加来表示任意向量。基向量的选取是线性代数的重要概念。

💡标量

标量指的是只有大小、没有方向的量。视频中2、3、-1、4等常数就是标量。标量可以缩放向量以改变向量大小,但不改变方向。标量与向量计算的引入,为后续线性代数打下了基础。

💡尾部和头部

向量箭头的一端叫做向量的尾部,另一端叫做向量的头部。视频中多次提到"从向量的尾部到头部"来描述向量。通过将一个向量的尾部和另一个向量的头部连接,可以实现向量的加法运算,这就是所谓的"向量三角形法则"。

Highlights

We can visually represent a vector as an arrow, where the length of the arrow is the magnitude of the vector and the direction of the arrow is the direction of the vector.

We can represent a vector mathematically by its components in the horizontal and vertical directions.

We can represent a vector as an ordered pair or 2-tuple, e.g., (2, 3) represents the vector with a horizontal component of 2 and a vertical component of 3.

Unit vectors i and j are introduced, where i represents the unit vector in the horizontal direction and j represents the unit vector in the vertical direction.

Any vector can be represented as a linear combination of the unit vectors i and j, e.g., v = 2i + 3j.

The vector v = (2, 3) can be written as 2i + 3j, where 2i represents the horizontal component and 3j represents the vertical component.

Unit vector notation allows for vector operations like addition and scaling.

To add two vectors in unit vector notation, add the corresponding components of each unit vector.

The vector addition v + b is demonstrated using unit vector notation and component addition.

The resulting vector v + b = i + 7j is equivalent to the column vector addition (2, 3) + (-1, 4) = (1, 7).

Unit vector notation provides a way to represent and manipulate vectors using their components and unit vectors.

Transcripts

We've already seen that you can visually

represent a vector as an arrow, where the length of the arrow

is the magnitude of the vector and the direction of the arrow

is the direction of the vector.

And if we want to represent this mathematically,

we could just think about, well, starting

from the tail of the vector, how far

away is the head of the vector in the horizontal direction?

And how far away is it in the vertical direction?

So for example, in the horizontal direction,

you would have to go this distance.

And then in the vertical direction,

you would have to go this distance.

Let me do that in a different color.

You would have to go this distance right over here.

And so let's just say that this distance is 2

and that this distance is 3.

We could represent this vector-- and let's

call this vector v. We could represent vector

v as an ordered list or a 2-tuple of-- so we could say we

move 2 in the horizontal direction

and 3 in the vertical direction.

So you could represent it like that.

You could represent vector v like this,

where it is 2 comma 3, like that.

And what I now want to introduce you to--

and we could come up with other ways of representing

this 2-tuple-- is another notation.

And this really comes out of the idea

of what it means to add and scale vectors.

And to do that, we're going to define

what we call unit vectors.

And if we're in two dimensions, we

define a unit vector for each of the dimensions

we're operating in.

If we're in three dimensions, we would define a unit vector

for each of the three dimensions that we're operating in.

And so let's do that.

So let's define a unit vector i.

And the way that we denote that is the unit vector

is, instead of putting an arrow on top,

we put this hat on top of it.

So the unit vector i, if we wanted

to write it in this notation right over here,

we would say it only goes 1 unit in the horizontal direction,

and it doesn't go at all in the vertical direction.

So it would look something like this.

That is the unit vector i.

And then we can define another unit vector.

And let's call that unit vector--

or it's typically called j, which

would go only in the vertical direction and not

in the horizontal direction.

And not in the horizontal direction,

and it goes 1 unit in the vertical direction.

So this went 1 unit in the horizontal.

And now j is going to go 1 unit in the vertical.

So j-- just like that.

Now any vector, any two dimensional vector,

we can now represent as a sum of scaled up versions of i and j.

And you say, well, how do we do that?

Well, you could imagine vector v right here

is the sum of a vector that moves purely

in the horizontal direction that has a length 2,

and a vector that moves purely in the vertical direction that

has length 3.

So we could say that vector v-- let

me do it in that same blue color--

is equal to-- so if we want a vector that has length 2

and it moves purely in the horizontal direction,

well, we could just scale up the unit vector i.

We could just multiply 2 times i.

So let's do that-- is equal to 2 times our unit vector i.

So 2i is going to be this whole thing

right over here or this whole vector.

Let me do it in this yellow color.

This vector right over here, you could view as 2i.

And then to that, we're going to add 3 times j-- so plus 3 times

j.

Let me write it like this.

Let me get that color.

Once again, 3 times j is going to be

this vector right over here.

And if you add this yellow vector right over here

to the magenta vector, you're going to get-- notice,

we're putting the tail of the magenta vector

at the head of the yellow vector.

And if you start at the tail of the yellow vector

and you go all the way to the head of the magenta vector,

you have now constructed vector v. So vector v,

you could represent it as a column vector like this, 2 3.

You could represent it as 2 comma 3,

or you could represent it as 2 times i with this little hat

over it, plus 3 times j, with this little hat over it.

i is the unit vector in the horizontal direction,

in the positive horizontal direction.

If you want to go the other way, you

would multiply it by a negative.

And j is the unit vector in the vertical direction.

As we'll see in future videos, once you

go to three dimensions, you'll introduce a k.

But it's very natural to translate between these two

things.

Notice, 2, 3-- 2, 3.

And so with that, let's actually do some vector operations

using this notation.

So let's say that I define another vector.

Let's say it is vector b.

I'll just come up with some numbers here.

Vector b is equal to negative 1 times i-- times the unit vector

i-- plus 4 times the unit vector in the horizontal direction.

So given these two vector definitions,

what would the would be the vector v plus b be equal to?

And I encourage you to pause the video and think about it.

Well once again, we just literally

have to add corresponding components.

We could say, OK, well let's think

about what we're doing in the horizontal direction.

We're going 2 in the horizontal direction here,

and now we're going negative 1.

So our horizontal component is going

to be 2 plus negative 1-- 2 plus negative 1

in the horizontal direction.

And we're going to multiply that times the unit vector i.

And this, once again, just goes back

to adding the corresponding components of the vector.

And then we're going to have plus 4, or plus 3 plus 4-- And

let me write it that way-- times the unit vector

j in the vertical direction.

And so that's going to give us-- I'll

do this all in this one color-- 2 plus negative 1 is 1i.

And we could literally write that just as i.

Actually, let's do that.

Let's just write that as i.

But we got that from 2 plus negative 1 is 1.

1 times the vector is just going to be that vector,

plus 3 plus 4 is 7-- 7j.

And you see, this is exactly how we saw vector addition

in the past, is that we could also represent vector

b like this.

We could represent it like this-- negative 1, 4.

And so if you were to add v to b,

you add the corresponding terms.

So if we were to add corresponding terms, looking

at them as column vectors, that is

going to be equal to 2 plus negative 1, which is 1.

3 plus 4 is 7.

So this is the exact same representation as this.

This is using unit vector notation,

and this is representing it as a column vector.