However, let's use E dA cosine theta instead of the dot product.

但是，让我们 E dA 余弦西塔来代替点。

Electric flux equals the integral of the dot product of electric field and dA.

等于场与dA的点的分。

In the example we were looking at, dot products certainly aren't preserved.

在我们看到的示例中，点当然会被保留。

Rather than using the dot product equation, let's use the electric flux equation  without the dot product.

我们点方程，而是带点的方程。

Electric flux equals the dot product of the electric field and the area, so let's  use that.

等于场与面的点，所以我们就它。

But the operation as a whole is not just one dot product but many.

但整个操作仅仅是一个点，而是多个点。

In fact, transformations which do preserve dot products are special enough to have their own name: Orthonormal transformations.

事实上，保留点的变换非常特殊，有自己的名字：正交变换。

Some of you might like think of this as a kind of  dot product.

你们中的一些人可能喜欢将其视为一种点。

In fact, worthwhile side note here transformations which do preserve dot products are special enough to have their own name.

事实上 值得注意的是保留点的变换很特别 有自己的名字。

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做点也就是叉乘。

Learn what you have learned, and imagine that you don't already know that the dot product relates to projection.

学习你学过的知识 想象一下你还知道点和投影的关系。

So when two vectors are generally pointing in the same direction, their dot product is positive.

所以当两个向常指向同一个方向时 它们的点是正的。

Notice, this looks like a dot product between two column vectors, [m1, m2], and [v1, v2].

请注意，这看起来像是两个列向 [m1, m2] 和 [v1, v2] 之间的点。

And if they point in generally the opposite direction, their dot product is negative.

如果它们常指向相反的方向 它们的点是负的。

Solving a linear system with an orthonormal matrix is actually super easy, because dot products are preserved.

标准正交矩阵求解线性方程组其实非常简单 因为点被保留了。

The dot product before and after the transformation will look very different.

变换前后的点看起来很一样。

When their perpendicular meaning, the projection of one onto the other, is the zero vector, their dot product is zero.

当它们垂直的意思 一个向在另一个向上的投影 是零向时 它们的点是零。

Luckily, this computation has a really nice geometric interpretation to think about the dot product between two vectors V and W.

幸运的是 这个计算有一个很好的几何解释来考虑两个向V和W之间的点。

For most linear transformations, the dot product before and after the transformation will be very different.

对于大多数线性变换，变换前后的点会有很大同。

That is, they don't preserve that zero dot product.

也就是说 它们保留0点。