Showing both the standard introduction to the topic,

同時給出對這個題材的標準引入

as well as a deeper view of how it relates to linear transformations.

以及更深入的它怎樣和線性轉換有關係。

I'd like to do the same thing for cross products,

我想對(叉積)外積也這樣做，

which also have a standard introduction

他也有一個標準的介紹說明

along with a deeper understanding in the light of linear transformations.

但也可以從線性轉換的角度更深入理解

But this time I am dividing it into two separate videos.

在這次的影片中，我將分成兩段影片分別說明

Here i'll try to hit the main points

我會盡力集中要點

that students are usually shown about the cross product.

說明學生通常如何被教導十字交乘法

And in the next video,

而在接下來的影片中，

I'll be showing a view which is less commonly taught, but really satisfying when you learn

我將展示一個不太常出現在教學中，但學習起來讓人很滿意的觀點

it.

我們從二維空間開始說起

We'll start in two dimensions.

如果你擁有兩個向量，v̅ 和w̅

If you have two vectors v̅ and w̅,

想像他們張開成一個平行四邊形

think about the parallelogram that they span out

我的意思是

What i mean by that is,

如果你複製一個v̅

that if you take a copy of v̅

將他從w̅的尾端移到w̅的頂端

and move its tail to the tip of w̅,

然後再複製一個w̅

and you take a copy of w̅

從v̅ 的尾端移到v̅ 的頂端

And move its tail to the tip of v̅,

四個向量在螢幕上圍成一個平行四邊形

the four vectors now on the screen enclose a certain parallelogram.

v̅ 和w̅的叉積(外積)

The cross product of v̅ and w̅,

可以用X形狀的乘法符號表示

written with the X-shaped multiplication symbol,

這是平行四邊形的面積

is the area of this parallelogram.

well , 大概是這樣。我們還需要考慮定向

Well, almost. We also need to consider

基本上，如果v̅ 在w̅右邊

orientation. Basically, if v̅ is on the

那v̅ X w̅ 是正值

right of w̅, then v̅×w̅ is positive

而且等於這個平行四邊形的面積

and equal to the area of the

但是，如果v̅ 在w̅左邊

parallelogram. But if v̅ is on the left of w̅,

那麼叉積(外積)為負數

then the cross product is negative,

也就是平行四邊形面積的負值

namely the negative area of that

注意，這表示外積相乘的順序很重要

parallelogram. Notice this means that

如果你交換 v̅ 和 w̅ 的位置

order matters. If you swapped v̅ and w̅

改成計算 w̅ X v̅

instead taking w̅×v̅, the cross

那麼 w̅ X v̅ 的叉積(外積)將變成負數

product would become the negative of

不管它之前是怎樣

whatever it was before. The way I always

我總是用這樣的方法來記住這裡的順序

remember the ordering here is that when

當你按照順序求兩個基本向量 î X ĵ 的叉積(外積)時

you take the cross product of the two

計算結果應該是正數。事實上，

basis vectors in order, î×ĵ,

你的基本向量的順序是什麼決定定向

the results should be positive. In fact,

所以因為 î 在 ĵ 右邊

the order of your basis vectors is what

我記得 v̅×w̅ 必須為正數

defines orientation so since î is on

當 v̅ 在w̅的右邊

the right of ĵ, I remember that v̅×w̅

所以，以影片所示的向量為例

has to be positive whenever v̅ is

我會跟你說這個平行四邊形的面積等於7

on the right of w̅.

而且因為 v̅ 在w̅左邊

So, for example with the vector shown

叉積(外積)應該是負的，因此v̅ X w̅ 為-7

here, I'll just tell you that the area of

但是當然地，你想在別人沒有告訴你面積等於多少的前提

that parallelogram is 7. And since v̅

計算出面積等於多少

is on the left of w̅, the cross product

這是行列式上場的時候了

should be negative so v̅×w̅ is -7.

所以，如果你沒有看過這個系列第5章討論行列式的部分

But of course you want to be able to

現在是一個很好的機會去看看

compute this without someone telling you

即使你看過，但已經過了一段時間

the area. This is where the determinant comes in.

我推薦你再看一次

So, if you didn't see Chapter 5 of this

確保這些想法在你的頭腦中仍然新鮮

series, where I talk about the

對於2-維的叉積 v̅×w̅，

determinant now would be a really good

你要做的就是你寫下v̅ 坐標

time to go take a look.

作為矩陣的第一個列

Even if you did see it, but it was a while

你走取w̅的坐標

ago. I'd recommend taking another look

將它們作爲第二個列然後你

just to make sure those ideas are fresh in your mind.

就計算行列式值。

For the 2-D cross-product v̅×w̅,

這是因為一個矩陣其列

what you do is you write the coordinates

表示V和W與對應

of v̅ as the first column of the matrix

移動所述線性變換

and you take the coordinates of w̅ and

基底矢量i和j，以v和w。

make them the second column then you

該決定是所有關於量度

just compute the determinant.

由於轉換型面積怎樣改變。

This is because a matrix whose columns

而且，我們期待原型區域

represent v̅ and w̅ corresponds with a

在是單位平方擱在i和j。

linear transformation that moves the

改造後，

basis vectors î and ĵ to v̅ and w̅.

該廣場被闢為

The determinant is all about measuring

我們關心的平行四邊形。

how areas change due to a transformation.

因此，決定通常這

And the prototypical area that we look

測量由哪些領域的因素

at is the unit square resting on î and ĵ.

改變，使這一區域

After the transformation,

平行四邊形;因為它從一個演變

that square gets turned into the

方與面積1日開始。

parallelogram that we care about.

如果v是在W的左邊更重要的是， 它

So the determinant which generally

意味著取向翻轉

measures the factor by which areas are

這種轉變，也就是在

changed, gives the area of this

它裝置，用於行列式為負。

parallelogram; since it evolved from a

舉個例子，讓我們講訴了

square that started with area 1.

坐標負（-3,1）和W具有

What's more if v̅ is on the left of w̅, it

坐標為（2,1）。的行列式

means that orientation was flipped

矩陣與這些坐標列

during that transformation, which is what

是（-3·1） - （2·1），

it means for the determinant to be negative.

這是-5。所以看樣子

As an example let's say v̅ has

我們定義的平行四邊形的面積為5

coordinates negative (-3,1) and w̅ has

並且因為V是W的左邊，它

coordinates (2,1). The determinant of the

應該有意義，這個值是

matrix with those coordinates as columns

負。正如你學習任何新的操作

is (-3·1) - (2·1),

我建議你玩弄此

which is -5. So evidently the

在你的頭它的概念只是為了讓

area of the parallelogram we define is 5

那種為了什麼直觀的感受

and since v̅ is on the left of w̅, it

跨產品的全部。

should make sense that this value is

例如，你可能會注意到，當

negative. As with any new operation you learn

兩個矢量是垂直的或在

I'd recommend playing around with this

至少接近垂直其

notion of it in your head just to get

叉積大於這將是

kind of an intuitive feel for what the

如果他們在非常相似的指向

cross product is all about.

方向。由於該地區

For example you might notice that when

平行四邊形較大時，雙方

two vectors are perpendicular or at

更接近垂直。

least close to being perpendicular their

別的東西，你可能會注意到的是，

cross product is larger than it would be

如果你要擴大其中的一個

if they were pointing in very similar

矢量，或許乘以V通過3

directions. Because the area of that

那麼平行四邊形的面積是

parallelogram is larger when the sides

也是由三個因素放大。

are closer to being perpendicular.

那麼是什麼意思這對於操作

Something else you might notice is that

該3V×寬將正好是三

if you were to scale up one of those

倍V×W的值。

vectors, perhaps multiplying v̅ by three

現在，儘管這一切的是一個

then the area of that parallelogram is

完全正常的數學運算

also scaled up by a factor of three.

我剛才所描述的是技術上不

So what this means for the operation is

交產物。真正的跨產品

that 3v̅×w̅ will be exactly three

是什麼，結合了兩種不同的

times the value of v̅×w̅ .

3D矢量得到一個新的三維矢量。就像 之前，

Now, even though all of this is a

我們仍然要考慮

perfectly fine mathematical operation

由兩個向量定義的平行四邊形

what i just described is technically not

這是一起穿越。和地區

the cross-product. The true cross product

這個平行四邊形的仍然是要

is something that combines two different

發揮很大的作用。為了具體的假設

3D vectors to get a new 3D vector. Just as before,

該區域是2.5的矢量

we're still going to consider the

這裡顯示但正如我所說的交

parallelogram defined by the two vectors

產品是不是一個數字它是一個載體。

that were crossing together. And the area

這個新的向量的長度將是區域

of this parallelogram is still going to

該平行四邊形，在這種情況下

play a big role. To be concrete let's say

2.5。而這新的方向

that the area is 2.5 for the vectors

矢量將是垂直於

shown here but as I said the cross

平行四邊形。但是，哪一種方式！對不對？

product is not a number it's a vector.

我的意思是有兩種可能的矢量

This new vector's length will be the area

長度2.5的垂直於給定的 平面。

of that parallelogram which in this case

這就是右手法則來

is 2.5. And the direction of that new

英寸把你的右手食指

vector is going to be perpendicular to

在V的方向，然後伸出

the parallelogram. But which way!, right?

中指在W的方向。

I mean there are two possible vectors with

然後，當你點了你的大拇指，這是 該

length 2.5 that are perpendicular to a given plane.

叉積的方向。

This is where the right hand rule comes

例如，讓我們說，五世一個

in. Put the fore finger of your right hand

長度為2指點直矢量

in the direction of v̅ then stick out

向上在Z方向上，而W是一個載體

your middle finger in the direction of w̅.

與純的Y軸長度2指點

Then when you point up your thumb, that's the

方向。平行四邊形，它們

direction of the cross product.

定義這個簡單的例子是

For example let's say that v̅ was a

實際上是一個正方形，因為他們是

vector with length 2 pointing straight

垂直，並且具有相同的長度。

up in the Z direction and w̅ is a vector

和正方形的面積是4。所以

with length 2 pointing in the pure Y

它們的橫產物應該是一個矢量

direction. The parallelogram that they

與長度4.使用右手

define in this simple example is

通常，他們的跨產品應指向 負X方向。

actually a square, since they're

所以這兩者的積

perpendicular and have the same length.

載體是-4·I。

And the area of that square is 4. So

對於更一般的計算，

their cross product should be a vector

有一個公式，你可以

with length 4. Using the right hand

記住，如果你想要，但它很常見

rule, their cross product should point in the negative X direction.

也更容易記住，而不是一定

So the cross product of these two

過程涉及到3D的決定因素。

vectors is -4·î.

現在，這個過程看起來確實奇怪

For more general computations,

第一。你寫下一個三維矩陣，其中

there is a formula that you could

在第二和第三列包含

memorize if you wanted but it's common

坐標V和W的。但對於

and easier to instead remember a certain

第一列你寫的基向量

process involving the 3D determinant.

I，J和k。然後你計算

Now, this process looks truly strange at

這個矩陣的行列式。該

first. You write down a 3D matrix where

愚蠢可能是在這裡清楚。

the second and third columns contain the

究竟是什麼意思放在一個

coordinates of v̅ and w̅. But for that

載體作為基體的條目？

first column you write the basis vectors

學生常常被告知，這是

î, ĵ and k̂. Then you compute

只是符號的把戲。當你隨身攜帶

the determinant of this matrix. The

出的計算，就好像I，J和K

silliness is probably clear here.

是數字，那麼你得到一些

What on earth does it mean to put in a

這些基向量的線性組合。

vector as the entry of a matrix?

和矢量

Students are often told that this is

由線性組合，學生定義

just a notational trick. When you carry

被告知只相信，是唯一

out the computations as if î, ĵ and k̂

矢量垂直於V和W，其

were numbers, then you get some

幅度是適當的區域

linear combination of those basis vectors.

平行四邊形，其方向服從

And the vector

右手法則。

defined by that linear combination, students

而且，一定！從某種意義上說，這只是一個

are told to just believe, is the unique

記法把戲。但是有一個理由

vector perpendicular to v̅ and w̅ whose

在做。

magnitude is the area of the appropriate

這不只是一個巧合

parallelogram and whose direction obeys

決定再次重要。和

the right hand rule.

把基本向量在這些時隙

And, sure!. In some sense this is just a

不只是做一個隨機的事情。至

notational trick. But there is a reason

明白的地方這一切來源於

for doing in.

它有助於使用二元的想法，

It's not just a coincidence that the

我在過去的視頻介紹。

determinant is once again important. And

這個概念是有點重

putting the basis vectors in those slots

雖然，所以我把它放在一個單獨

is not just a random thing to do. To

後續視頻任你是誰

understand where all of this comes from

好奇的了解更多。

it helps to use the idea of duality that

可以說，它屬於精髓外

I introduced in the last video.

線性代數。這裡最重要的組成部分

This concept is a little bit heavy

要知道這是什麼叉積

though, so I'm putting it in a separate

矢量幾何表示。因此，如果

follow-on video for any of you who are

你想跳過下一個視頻，感覺

curious to learn more.

自由。但是，對於那些你們誰是

Arguably it falls outside the essence of

願意去深一點，誰是

linear algebra. The important part here

好奇之間的連接

is to know what that cross product

這種計算和底層

vector geometrically represents. So if

幾何形狀，想法，我就說說

you want to skip that next video, feel

在接下來的視頻或只是一個真

free. But for those of you who are

優雅的一塊數學。

willing to go a bit deeper and who are

curious about the connection between

this computation and the underlying

geometry, the ideas that I will talk about

in the next video or just a really

elegant piece of math.