Square of Norm of Vector Cross Product

Theorem
Let $\mathbf a$ and $\mathbf b$ be vectors in the Euclidean space $\R^3$.

Let $\times$ denote the vector cross product.

Then:


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

Proof
Let $\mathbf a = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}$, and $\mathbf b = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$.

Then: