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

但，让我们使用 E dA 余弦西塔来代替点。

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

通量等于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，点当然不会被保留。

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

我们不使用点方程，使用不带点的通量方程。

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

通量等于面的点，所以我们就使用它。

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

但整个操作不仅仅一个点，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量做点也就叉乘。

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 想象一下你还不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.

所以当两个向量通常指向同一个方向时 它们的点正的。

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

请注意，这看起来像两个列向量 [m1, m2] 和 [v1, v2] 之间的点。

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

如果它们通常指向相反的方向 它们的点负的。

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

用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不一样。

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 一个向量在另一个向量上的投影 零向量时 它们的点零。

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

幸运的 这个计算有一个很好的几何解释来考虑两个向量V和W之间的点。

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

对于大数线性变换，变换前后的点会有很大不同。

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

也就说 它们不保留0点。