The actual 3D cross product takes in two vectors and spits out a vector.

实际的三维外积是两个向量 然后得到一个向量。

But this idea actually gets us really close to what the real cross product is.

但是这个想法实际上让我们非常接近真正的叉乘。

So that performing the linear transformation is the same as taking a dot product with that vector the cross product.

所以进行线性变换就等于对这个向量做点积也就是叉乘。

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

在这里 我将试着抓住学生们通常学到的关于叉乘的要点。

Of course, you all know that this is not the 3D cross product.

当然 你们都知道这不是三维叉乘。

This is the fundamental reason why the computation and the geometric interpretation of the cross product are related.

这就是为什么叉乘的计算和几何解释是相关的根本原因。

So that wraps up dot products and cross products.

这就包含了点积和叉乘。

The cross product of VNW, written with the X shaped multiplication symbol, is the area of this parallelogram.

VNW的叉乘 用X形乘法符号表示 就是这个平行四边形的面积。

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

这里重要的部分是知道叉乘向量在几何上表示什么。

So to back up a bit, remember in two dimensions what it meant to compute the two D version of the cross product.

回想一下 在二维空间中 计算二维空间的外积意味着什么。

Using the right hand rule, their cross products should point in the negative X direction, so the cross product of these two vectors is negative four times.

用右手定则 它们的叉乘应该指向-X方向 所以这两个向量的叉乘是-4倍。

But if V is on the left of W, then the cross product is negative, namely, the negative area of that parallelogram.

但是如果V在W的左边 那么叉乘就是负的 也就是平行四边形的负面积。

If you swapped V-N-W, instead taking W, cross V, the cross product would become the negative of whatever it was before.

如果交换V-n-W 而是用W叉乘V 叉乘就变成了之前的负数。

Then when we associate that transformation with its dual vector in three D space, that dual vector is going to be the cross product of V-N-W.

然后当我们把这个变换和它在三维空间中的对偶向量联系起来 这个对偶向量就是V-N-W的外积。

The reason for doing this will be that understanding that transformation is gonna make clear the connection between the computation and the geometry of the cross product.

这样做的原因是理解这个变换会让计算和叉乘的几何之间的联系更加清晰。

Since they're perpendicular and have the same length, and the area of that square is four, so their cross product should be a vector with length four.

因为它们互相垂直 长度相等 这个正方形的面积是4 所以它们的叉乘应该是一个长度为4的向量。

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.

我建议大家在脑子里玩一下这个概念 只是为了对叉乘有个直观的感觉。

Point the forefinger of your right hand in the direction of V, then stick out your middle finger in the direction of W, then when you point up your thumb, that's the direction of the cross product.

右手的食指指向V的方向 然后伸出中指指向W的方向 然后当你的拇指向上指向时 这就是叉乘的方向。