Norm of Vector Cross Product

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

Let $\times$ denote the vector cross product.

Then:


 * $\norm {\mathbf a \times \mathbf b} = \norm {\mathbf a} \norm {\mathbf b} \size {\sin \theta}$

where $\theta$ is the angle between $\mathbf a$ and $\mathbf b$, or an arbitrary number if $\mathbf a$ or $\mathbf b$ is the zero vector.

Proof
Suppose either $\mathbf a$ or $\mathbf b$ is the zero vector.

Then by :
 * $\norm {\mathbf a} = 0$

or:
 * $\norm {\mathbf b} = 0$

By calculation, it follows that $\mathbf a \times \mathbf b$ is also the zero vector.

Hence:
 * $\norm {\mathbf a \times \mathbf b} = 0$

and equality holds.

Now suppose that both $\mathbf a$ or $\mathbf b$ are non-zero vectors.

We have: