Since both of these approaches give us a dual vector to the same transformation, they must be the same vector.

因为这两种方法都给出了同一个变换的对偶向量 所以它们一定是相同的向量。

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。

But thinking geometrically, we can deduce that this dual vector must be perpendicular to VMW with a length equal to the area of the parallelogram spanned out by those two vectors.

但从几何角度考虑 我们可以推导出这个对偶向量必须垂直于VMW 其长度等于由这两个向量张成的平行四边形的面积。

The takeaway is that whenever you're out in the mathematical wild and you find a linear transformation to the number line, you will be able to match it to some vector, which is called the dual vector of that transformation.

只要你在数学领域中找到一个到数轴上的线性变换 你就能将它与某个向量相匹配 这个向量被称为这个变换的对偶向量。