## 字幕列表 影片播放

• Last video, I've talked about the dot product.

上一個視頻，我已經談到了(點積)內積

• 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 vectorsand 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

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

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 ofand 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, ifis 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 ifis 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 swappedand

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 sinceis 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 wheneveris

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

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

• ofas the first column of the matrix

移動所述線性變換

• and you take the coordinates ofand

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

• representandcorresponds with a

在是單位平方擱在i和j。

• linear transformation that moves the

改造後，

• basis vectorsandtoand 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 onand 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 ifis 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 sayhas

我們定義的平行四邊形的面積為5

• coordinates negative (-3,1) andhas

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 sinceis 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

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 multiplyingby 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 ofthen 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 thatwas 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 andis 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 ofand 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

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 toandwhose

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

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.

Last video, I've talked about the dot product.

B1 中級 中文 美國腔 面積 矩陣 線性 計算 長度 方向

# 3blue1brown 線性代數精髓系列第 10 章 - Cross products 外積/叉積（Cross products | Essence of linear algebra, Chapter 10）

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tai 發佈於 2021 年 02 月 16 日