外積 | 第10章，線性代數的本質：英語精讀講義

Last video, I talked about the dot product,

在上個影片中，我談到了內積，

showing both the standard introduction to the topic, 文法

既展示了該主題的標準入門介紹，

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

也深入探討了它與線性轉換的關係。

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

我想對外積做同樣的事情，

which also have a standard introduction

它同樣有一個標準的入門介紹，

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

以及從線性轉換角度出發的更深入理解。

But this time, I'm dividing it into two separate videos.

但這次，我把它分成了兩個單獨的影片。

Here, I'll try to hit the main points that students are usually shown about the cross product. 慣用語

在這裡，我將盡量講清學生們通常學到的有關外積的主要知識點。

And in the next video,

而在下個影片裡，

I'll be showing a view which is less commonly taught,

我將展示一種較少被教授的視角，

but really satisfying when you learn it.

但當你學會後會覺得非常滿足。

We'll start in two dimensions.

我們先從二維開始。

If you have two vectors V and W,

如果你有兩個向量 V 和 W，

think about the parallelogram that they span out. 片語動詞

想像一下它們所張成的平行四邊形。

What I mean by that is that if you take a copy of V and move its tail to the tip of W, 講解

我的意思是，如果你把 V 複製一份，將其尾部移到 W 的尖端，

and you take a copy of W and move its tail to the tip of V,

再把 W 複製一份，將其尾部移到 V 的尖端，

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

螢幕上的這四個向量就會圍成一個特定的平行四邊形。

The cross product of V and W,

V 和 W 的外積，

written with the X-shaped multiplication symbol,

用一個 X 形的乘號來表示，

is the area of this parallelogram.

就是這個平行四邊形的面積。

Well, almost. 表達脈絡

嗯，差不多是這樣。

We also need to consider orientation.

我們還需要考慮方向性。

Basically, if V is on the right of W,

基本上，如果 V 在 W 的右側，

then V cross W is positive and equal to the area of the parallelogram.

那麼 V 外積 W 是正的，且等於該平行四邊形的面積。

But if V is on the left of W,

但如果 V 在 W 的左側，

then the cross product is negative,

那麼外積就是負的，

namely the negative area of that parallelogram.

也就是那個平行四邊形面積的負值。

Notice, this means that order matters. 片語搭配

注意，這意味著順序很重要。

If you swapped V and W,

如果你交換了 V 和 W，

instead taking W cross V,

變成計算 W 外積 V，

the cross product would become the negative of whatever it was before. 文法

外積就會變成之前結果的相反數。

The way I always remember the ordering here is that when you take the cross product of the two basis vectors in order,

我記住這裡順序的方法是，當你按順序對兩個基底向量求外積時，

i-hat cross j-hat, the result should be positive.

i帽外積 j帽，結果應該是正的。

In fact, the order of your basis vectors is what defines orientation.

實際上，你基底向量的順序就是定義方向的標準。

So, since i-hat is on the right of j-hat,

所以，既然 i帽 在 j帽 的右側，

I remember that V cross W has to be positive whenever V is on the right of W.

我就記住了，只要 V 在 W 的右側，V 外積 W 就必須是正的。

So, for example, with the vectors shown here,

那麼，舉個例子，就拿這裡顯示的向量來說，

I'll just tell you that the area of that parallelogram is seven.

我直接告訴你那個平行四邊形的面積是 7。

And since V is on the left of W,

由於 V 在 W 的左側，

the cross product should be negative.

外積應該是負的。

So, V cross W is negative seven.

所以，V 外積 W 等於 -7。

But of course, you want to be able to compute this without someone telling you the area.

當然，你肯定希望在沒人告訴你面積的情況下也能計算出這個值。

This is where the determinant comes in. 片語動詞

這就是行列式派上用場的地方了。

So, if you didn't see chapter five of this series, 文法

所以，如果你還沒看過這個系列的第五章，

where I talk about the determinant,

也就是我講行列式的那一期，

now would be a really good time to go take a look.

現在正是去看看的好時機。

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

即使你看過，但如果過了一段時間，

I'd recommend taking another look just to make sure those ideas are fresh in your mind. 慣用語

我建議你再複習一下，確保這些概念在你的腦海中依然清晰。

For the 2D cross product, V cross W,

對於二維外積，V 外積 W，

what you do is you write the coordinates of V as the first column of a matrix,

你的做法是將 V 的座標寫成矩陣的第一行（直行），

and you take the coordinates of W and make them the second column.

並把 W 的座標作為矩陣的第二行（直行）。

Then you just compute the determinant.

然後你只需計算該行列式。

This is because a matrix whose columns represent V and W corresponds with a linear transformation that moves the basis vectors i-hat and j-hat to V and W.

這是因為，一個以 V 和 W 為行的矩陣，對應於一個線性轉換，它將基底向量 i帽 和 j帽 移動到 V 和 W 的位置。

The determinant is all about measuring how areas change due to a transformation.

行列式的核心在於衡量面積是如何因轉換而改變的。

And the prototypical area that we look at is the unit square resting on i-hat and j-hat.

而我們研究的典型面積就是停在 i帽 和 j帽 上的單位正方形。

After the transformation,

在轉換之後，

that square gets turned into the parallelogram that we care about. 片語動詞

那個正方形變成了我們關注的平行四邊形。

So, the determinant, which generally measures the factor by which areas are changed, 文法

所以，這個通常用來衡量面積改變倍數的行列式，

gives the area of this parallelogram,

就給出了這個平行四邊形的面積，

since it evolved from a square that started with area one.

因為它是由一個初始面積為 1 的正方形演變而來的。

What's more, if V is on the left of W,

更重要的是，如果 V 在 W 的左側，

it means that orientation was flipped during that transformation,

這意味著方向在那個轉換過程中被翻轉了，

which is what it means for the determinant to be negative. 文法

這也就是行列式為負的意義所在。

As an example, let's say V has coordinates negative three one,

舉個例子，假設 V 的座標是負三和一，

and W has coordinates two one.

W 的座標是二和一。

The determinant of the matrix with those coordinates as columns is negative three times one minus two times one,

以這些座標為行的矩陣行列式是負三乘以一減去二乘以一，

which is negative five.

等於負五。

So, evidently, the area of the parallelogram they define is five.

所以很明顯，它們所定義的平行四邊形面積是五。

And since V is on the left of W,

且因為 V 在 W 的左側，

it should make sense that this value is negative.

這個值為負也就完全說得通了。

As with any new operation you learn,

就像你學習任何新的運算一樣，

I'd recommend playing around with this notion a bit in your head just to get kind of an intuitive feel for what the cross product is all about. 講解

我建議你在腦海中稍微把玩一下這個概念，以直觀地感受外積到底是怎麼回事。

For example, you might notice that when two vectors are perpendicular,

例如，你可能會注意到，當兩個向量垂直，

or at least close to being perpendicular,

或者至少接近垂直時，

their cross product is larger than it would be if they were pointing in very similar directions, 文法

它們的外積會比它們指向非常相似的方向時要大，

because the area of that parallelogram is larger when the sides are closer to being perpendicular.

因為當邊更接近垂直時，那個平行四邊形的面積更大。

Something else you might notice is that if you were to scale up one of those vectors, 片語動詞

你可能還會注意到的另一件事是，如果你放大其中一個向量，

perhaps multiplying V by three,

比如將 V 乘以 3，

then the area of that parallelogram is also scaled up by a factor of three. 片語搭配

那麼那個平行四邊形的面積也會按 3 的比例放大。

So, what this means for the operation is that three V cross W will be exactly three times the value of V cross W. 表達脈絡

所以，這對該運算意味著，3V 外積 W 的值，將正好是 V 外積 W 值的 3 倍。

Now, even though all of this is a perfectly fine mathematical operation, 文法

雖然這完全是一個合理的數學運算，

what I just described is technically not the cross product. 用詞

但我剛才描述的，嚴格來說並不是外積。

The true cross product is something that combines two different 3D vectors to get a new 3D vector.

真正的外積是結合兩個不同的 3D 向量來得到一個新的 3D 向量。

Just as before, 表達脈絡

和以前一樣，

we're still going to consider the parallelogram defined by the two vectors that we're crossing together.

我們仍然要考慮由我們要進行外積的兩個向量所張成的平行四邊形。

And the area of this parallelogram is still going to play a big role. 慣用語

而這個平行四邊形的面積，依然會扮演重要的角色。

To be concrete, let's say that the area is 2.5 for the vectors shown here. 表達脈絡

具體來說，假設這裡顯示的向量所圍成的面積是 2.5。

But as I said, the cross product is not a number,

但正如我所說，外積不是一個數字，

it's a vector.

它是一個向量。

This new vector's length will be the area of that parallelogram,

這個新向量的長度，將是那個平行四邊形的面積，

which in this case is 2.5.

在這個例子中是 2.5。

And the direction of that new vector is going to be perpendicular to the parallelogram.

而那個新向量的方向，將會垂直於該平行四邊形。

But which way, right?

但是，向哪個方向呢？

I mean, there are two possible vectors with length 2.5 that are perpendicular to a given plane.

我的意思是，垂直於給定平面、長度為 2.5 的向量有兩個可能的方向。

This is where the right hand rule comes in. 用詞

這就是右手定則派上用場的地方了。

Point the forefinger of your right hand in the direction of V.

將你右手的食指指向 V 的方向。

Then stick out your middle finger in the direction of W. 片語動詞

然後伸出中指指向 W 的方向。

Then, when you point up your thumb,

接著，當你豎起大拇指時，

that's the direction of the cross product.

那就是外積的方向。

For example, let's say that V was a vector with length two pointing straight up in the Z direction,

例如，假設 V 是一個長度為 2、沿 Z 軸直指上方的向量，

and W is a vector with length two pointing in the pure Y direction.

而 W 是一個長度為 2、指向純 Y 軸方向的向量。

The parallelogram that they define in this simple example is actually a square,

在這個簡單的例子中，它們定義的平行四邊形實際上是一個正方形，

since they're perpendicular and have the same length. 文法

因為它們互相垂直且長度相同。

And the area of that square is four.

而那個正方形的面積是 4。

So, their cross product should be a vector with length four.

所以，它們的外積應該是一個長度為 4 的向量。

Using the right hand rule,

使用右手定則，

their cross product should point in the negative X direction.

它們的外積應該指向 X 軸的負方向。

So, the cross product of these two vectors is negative four times i-hat.

所以，這兩個向量的外積是負四倍的 i帽。

For more general computations,

對於更一般的計算，

there is a formula that you could memorize if you wanted, 語氣意圖

如果你願意，有一個公式可以背下來，

but it's common and easier to instead remember a certain process involving the 3D determinant.

但通常更容易記住的，是一個涉及 3D 行列式的特定過程。

Now, this process looks truly strange at first.

現在，這個過程乍一看真的很奇怪。

You write down a 3D matrix where the second and third columns contain the coordinates of V and W.

你寫下一個 3D 矩陣，其中第二和第三行（直行）包含 V 和 W 的座標。

But for that first column,

但對於第一行，

you write the basis vectors i-hat, j-hat, and k-hat.

你要寫上基底向量 i帽、j帽 和 k帽。

Then, you compute the determinant of this matrix.

然後，你計算這個矩陣的行列式。

The silliness is probably clear here.

這種做法的荒謬之處大概很明顯。

What on earth does it mean to put in a vector as the entry of a matrix? 慣用語

把一個向量作為矩陣的元素填進去到底是什麼意思？

Students are often told that this is just a notational trick.

學生們經常被告知，這只是一個符號上的把戲。

When you carry out the computations as if i-hat, j-hat, and k-hat were numbers, 講解

當你像對待數字那樣對待 i帽、j帽 和 k帽 進行計算時，

then you get some linear combination of those basis vectors.

你就會得到這些基底向量的某個線性組合。

And the vector defined by that linear combination,

由那個線性組合定義的向量，

students are told to just believe, 語氣意圖

學生們被告知只需相信，

is the unique vector perpendicular to V and W whose magnitude is the area of the appropriate parallelogram

就是唯一垂直於 V 和 W 的向量，其大小等於相應平行四邊形的面積，

and whose direction obeys the right hand rule.

且其方向遵循右手定則。

And sure, in some sense this is just a notational trick. 語氣意圖

確實，在某種意義上，這只是一個符號上的把戲。

But there is a reason for doing it.

但這樣做是有原因的。

It's not just a coincidence that the determinant is once again important. 表達脈絡

行列式再次變得重要，並非巧合。

And putting the basis vectors in those slots is not just a random thing to do.

而把基底向量放在這些位置上，也不是隨便做的。

To understand where all of this comes from,

為了理解這一切的由來，

it helps to use the idea of duality that I introduced in the last video.

我在上個影片中介紹的對偶性思想會很有幫助。

This concept is a little bit heavy, 慣用語

不過，這個概念稍微有點深奧。

though.

所以，我把它放在了單獨的後續影片中，供任何好奇想了解更多的朋友觀看。

So, I'm putting it in a separate follow-on video for any of you who are curious to learn more.

可以說，它超出了線性代數本質的範疇。

Arguably, it falls outside the essence of linear algebra. 講解

這裡重要的部分是知道那個外積向量在幾何上代表什麼。

The important part here is to know what that cross product vector geometrically represents.

所以，如果你想跳過下個影片，

So, if you want to skip that next video,

請隨意。

feel free. 語氣意圖

但對於那些願意更深入探索，並且對這種計算與底層幾何之間的聯繫感到好奇的人來說，

But for those of you who are willing to go a bit deeper and who are curious about the connection between this computation and the underlying geometry, 慣用語

我將在下個影片中討論的思想，真的是一段非常優雅的數學。

the ideas that I'll talk about in the next video are just a really elegant piece of math.

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