叉积 | 第10章，线性代数的本质：英语精读讲义

Last video, I talked about the dot product,

上个视频中，我讲了点积，

showing both the standard introduction to the topic, 語法

既展示了该主题的标准入门介绍，

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

也深入探讨了它与线性变换的关系。

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

我想对叉积做同样的事情，

which also have a standard introduction

它同样有一个标准的入门介绍，

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

以及从线性变换角度出发的更深入理解。

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

但这次，我把它分成了两个单独的视频。

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

在这里，我将尽量讲清学生们通常学到的有关叉积的主要知识点。

And in the next video,

而在下个视频里，

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

我将展示一种较少被教授的视角，

but really satisfying when you learn it.

但当你学会后会觉得非常满足。

We'll start in two dimensions.

我们先从二维开始。

If you have two vectors V and W,

如果你有两个向量 V 和 W，

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

想象一下它们所张成的平行四边形。

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 复制一份，将其尾部移到 W 的尖端，

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

再把 W 复制一份，将其尾部移到 V 的尖端，

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

屏幕上的这四个向量就会围成一个特定的平行四边形。

The cross product of V and W,

V 和 W 的叉积，

written with the X-shaped multiplication symbol,

用一个 X 形的乘号来表示，

is the area of this parallelogram.

就是这个平行四边形的面积。

Well, almost. 表達脈絡

嗯，差不多是这样。

We also need to consider orientation.

我们还需要考虑方向。

Basically, if V is on the right of W,

基本上，如果 V 在 W 的右侧，

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

那么 V 叉乘 W 是正的，且等于该平行四边形的面积。

But if V is on the left of W,

但如果 V 在 W 的左侧，

then the cross product is negative,

那么叉积就是负的，

namely the negative area of that parallelogram.

也就是那个平行四边形面积的负值。

Notice, this means that order matters. 短語搭配

注意，这意味着顺序很重要。

If you swapped V and W,

如果你交换了 V 和 W，

instead taking W cross V,

变成计算 W 叉乘 V，

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

叉积就会变成之前结果的相反数。

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

我记住这里顺序的方法是，当你按顺序对两个基向量求叉积时，

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

i帽叉乘 j帽，结果应该是正的。

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

实际上，你基向量的顺序就是定义方向的标准。

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

所以，既然 i帽 在 j帽 的右侧，

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

我就记住了，只要 V 在 W 的右侧，V 叉乘 W 就必须是正的。

So, for example, with the vectors shown here,

那么，举个例子，就拿这里显示的向量来说，

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

我直接告诉你那个平行四边形的面积是 7。

And since V is on the left of W,

由于 V 在 W 的左侧，

the cross product should be negative.

叉积应该是负的。

So, V cross W is negative seven.

所以，V 叉乘 W 等于 -7。

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

当然，你肯定希望在没人告诉你面积的情况下也能计算出这个值。

This is where the determinant comes in. 短語動詞

这就是行列式派上用场的地方了。

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

所以，如果你还没看过这个系列的第五章，

where I talk about the determinant,

也就是我讲行列式的那一期，

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

现在正是去看看的好时机。

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

即使你看过，但如果过了一段时间，

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

我建议你再复习一下，确保这些概念在你的脑海中依然清晰。

For the 2D cross product, V cross W,

对于二维叉积，V 叉乘 W，

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

你的做法是将 V 的坐标写成矩阵的第一列，

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

并把 W 的坐标作为矩阵的第二列。

Then you just compute the determinant.

然后你只需计算该行列式。

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.

这是因为，一个以 V 和 W 为列的矩阵，对应于一个线性变换，它将基向量 i帽 和 j帽 移动到 V 和 W 的位置。

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

行列式的核心在于衡量面积是如何因变换而改变的。

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

而我们研究的典型面积就是停在 i帽 和 j帽 上的单位正方形。

After the transformation,

在变换之后，

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

那个正方形变成了我们关注的平行四边形。

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

所以，这个通常用来衡量面积改变倍数的行列式，

gives the area of this parallelogram,

就给出了这个平行四边形的面积，

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

因为它是由一个初始面积为 1 的正方形演变而来的。

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

更重要的是，如果 V 在 W 的左侧，

it means that orientation was flipped during that transformation,

这意味着方向在那个变换过程中被翻转了，

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

这也就是行列式为负的意义所在。

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

举个例子，假设 V 的坐标是负三和一，

and W has coordinates two one.

W 的坐标是二和一。

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

以这些坐标为列的矩阵行列式是负三乘以一减去二乘以一，

which is negative five.

等于负五。

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

所以很明显，它们所定义的平行四边形面积是五。

And since V is on the left of W,

且因为 V 在 W 的左侧，

it should make sense that this value is negative.

这个值为负也就完全说得通了。

As with any new operation you learn,

就像你学习任何新的运算一样，

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

我建议你在脑海中稍微把玩一下这个概念，以直观地感受叉积到底是怎么回事。

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

例如，你可能会注意到，当两个向量垂直，

or at least close to being perpendicular,

或者至少接近垂直时，

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

它们的叉积会比它们指向非常相似的方向时要大，

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

因为当边更接近垂直时，那个平行四边形的面积更大。

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

你可能还会注意到的另一件事是，如果你放大其中一个向量，

perhaps multiplying V by three,

比如将 V 乘以 3，

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

那么那个平行四边形的面积也会按 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，这对该运算意味着，3V 叉乘 W 的值，将正好是 V 叉乘 W 值的 3 倍。

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

虽然这完全是一个合理的数学运算，

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

但我刚才描述的，严格来说并不是叉积。

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

真正的叉积是结合两个不同的 3D 向量来得到一个新的 3D 向量。

Just as before, 表達脈絡

和以前一样，

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

我们仍然要考虑由我们要进行叉乘的两个向量所张成的平行四边形。

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

而这个平行四边形的面积，依然会扮演重要的角色。

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

具体来说，假设这里显示的向量所围成的面积是 2.5。

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

但正如我所说，叉积不是一个数字，

it's a vector.

它是一个向量。

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

这个新向量的长度，将是那个平行四边形的面积，

which in this case is 2.5.

在这个例子中是 2.5。

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

而那个新向量的方向，将会垂直于该平行四边形。

But which way, right?

但是，向哪个方向呢？

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

我的意思是，垂直于给定平面、长度为 2.5 的向量有两个可能的方向。

This is where the right hand rule comes in. 用詞

这就是右手定则派上用场的地方了。

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

将你右手的食指指向 V 的方向。

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

然后伸出中指指向 W 的方向。

Then, when you point up your thumb,

接着，当你竖起大拇指时，

that's the direction of the cross product.

那就是叉积的方向。

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

例如，假设 V 是一个长度为 2、沿 Z 轴直指上方的向量，

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

而 W 是一个长度为 2、指向纯 Y 轴方向的向量。

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

在这个简单的例子中，它们定义的平行四边形实际上是一个正方形，

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

因为它们互相垂直且长度相同。

And the area of that square is four.

而那个正方形的面积是 4。

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

所以，它们的叉积应该是一个长度为 4 的向量。

Using the right hand rule,

使用右手定则，

their cross product should point in the negative X direction.

它们的叉积应该指向 X 轴的负方向。

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

所以，这两个向量的叉积是负四倍的 i帽。

For more general computations,

对于更一般的计算，

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

如果你愿意，有一个公式可以背下来，

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

但通常更容易记住的，是一个涉及 3D 行列式的特定过程。

Now, this process looks truly strange at first.

现在，这个过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.

你写下一个 3D 矩阵，其中第二和第三列包含 V 和 W 的坐标。

But for that first column,

但对于第一列，

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

你要写上基向量 i帽、j帽 和 k帽。

Then, you compute the determinant of this matrix.

然后，你计算这个矩阵的行列式。

The silliness is probably clear here.

这种做法的荒诞之处大概很明显。

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

把一个向量作为矩阵的元素填进去到底是什么意思？

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

学生们经常被告知，这只是一个符号上的把戏。

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

当你像对待数字那样对待 i帽、j帽 和 k帽 进行计算时，

then you get some linear combination of those basis vectors.

你就会得到这些基向量的某个线性组合。

And the vector defined by that linear combination,

由那个线性组合定义的向量，

students are told to just believe, 語氣意圖

学生们被告知只需相信，

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

就是唯一垂直于 V 和 W 的向量，其大小等于相应平行四边形的面积，

and whose direction obeys the right hand rule.

且其方向遵循右手定则。

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

确实，在某种意义上，这只是一个符号上的把戏。

But there is a reason for doing it.

但这样做是有原因的。

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

行列式再次变得重要，并非巧合。

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

而把基向量放在这些位置上，也不是随便做的。

To understand where all of this comes from,

为了理解这一切的由来，

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

我在上个视频中介绍的对偶性思想会很有帮助。

This concept is a little bit heavy, 習語

不过，这个概念稍微有点深奥。

though.

所以，我把它放在了单独的后续视频中，供任何好奇想了解更多的朋友观看。

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

可以说，它超出了线性代数本质的范畴。

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

这里重要的部分是知道那个叉积向量在几何上代表什么。

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

所以，如果你想跳过下个视频，

So, if you want to skip that next video,

请随意。

feel free. 語氣意圖

但对于那些愿意更深入探索，并且对这种计算与底层几何之间的联系感到好奇的人来说，

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, 習語

我将在下个视频中讨论的思想，真的是一段非常优雅的数学。

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

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