Square of Vector Cross Product

Theorem
Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $3$ dimensions.

Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$.

Then:


 * $\paren {\mathbf a \times \mathbf b}^2 = \mathbf a^2 \mathbf b^2 - \paren {\mathbf a \cdot \mathbf b}^2$

where:
 * $\paren {\mathbf a \times \mathbf b}^2$ denotes the square of $\mathbf a \times \mathbf b$
 * $\mathbf a \cdot \mathbf b$ denotes the dot product of $\mathbf a \times \mathbf b$.

Proof 2
The following proof is valid in the context of a real Euclidean space of $3$ dimensions: