Dot Product of Sum with Difference of Vectors

Theorem
Let $\mathbf a$ and $\mathbf b$ be vector quantities.

Then:
 * $\paren {\mathbf a + \mathbf b} \cdot \paren {\mathbf a - \mathbf b} = \mathbf a^2 - \mathbf b^2$

where:
 * $\cdot$ denotes dot product
 * $\mathbf a^2$ denotes the square of $\mathbf a$, that is: $\mathbf a \cdot \mathbf a$.

Also see

 * Square of Sum of Vectors
 * Difference of Two Squares