Cross Product of Antiparallel Vectors is Zero

Theorem
Let $\mathbf u$ and $\mathbf v$ be antiparallel vectors.

Then:
 * $\mathbf u \times \mathbf v = 0$

where $\times$ denotes vector cross product.

Proof
We have that $\mathbf u$ and $\mathbf v$ are antiparallel.

Hence we can define $\mathbf u$ and $\mathbf v$ as:

where $x$, $y$ and $z$ are arbitrary real numbers.

By definition of vector cross product:


 * $\mathbf u \times \mathbf v = \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ x & y & z \\ -x & -y & -z \end {vmatrix}$

It is seen that the $3$rd row is a multiple of the $2$nd row.

The result then follows from Determinant with Row Multiplied by Constant.