Dot Product of Antiparallel Vectors is Negative

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

Then:
 * $\mathbf u \cdot \mathbf v < 0$

where $\cdot$ denotes dot 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 such that at least one of $x$, $y$ and $z$ is not zero.

By definition of dot product: