Meaning our final output looks like that relevant dot product, but minus one.

意思是我们最终输出看起来像那个相关点，但减去一。

Otherwise, that dot product would be 0 or negative, meaning the vector doesn't really align with that direction.

否则， 那个点就会是0或负数，这意味着这个向量并没有真正那个方向对齐。

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

通量等于dA点。

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

通量等于点乘，所以我们使用这个公式。

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

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

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

然而，我们使用E dA 余弦 theta 而不是点乘。

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

如果它们通常指向相反方向 它们点是负。

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

其使用点乘方程，让我们使用不带点乘通量方程。

So that wraps up dot products and cross products.

这就包含了点和叉乘。

Similarly, its dot product with these other directions would tell you whether it represents the last name Jordan, or basketball.

类似地， 它这些其他方向点会告诉你它代表是姓乔丹，还是篮球。

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，变换前后点会非常不同。

And for simplicity, let's completely ignore the very reasonable question of what it might mean if that dot product was bigger than 1.

为了简化问题，我们完全忽略掉如果点大于1可能意味着什么这一合理问题。

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 它们点是正。

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

有些人可能会认为这类似于点。

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

事实上，保持点不变变换特殊到有自己名称：正交变换。

Now, this numerical operation of multiplying a one by two matrix by a vector feels just like taking the dot product of two vectors.

现在 这个1×2矩阵乘以一个向量数值运算就像取两个向量点。

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

但整个操作不仅仅是单一点，而是多个点。

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

幸运是 这个计算有一个很好几何解释来考虑两个向量V和W之间点。

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

当它们垂直意思 一个向量在另一个向量上投影 是零向量时 它们点是零。

Looking at sides 2 and 4, we need to realize the dot product of B and ds, is the same as B ds cosine theta, Right!

观察边2和边4， 我们需要实现B和ds点，B ds cosine theta相同， 对吧！