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:


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

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
If $\mathbf a$ or $\mathbf b$ is the zero vector, then $\norm{\mathbf a} = 0$, or $\norm{\mathbf b}= 0$ by.

By calculation, it follows that $\mathbf a \times \mathbf b$ is also the zero vector, so $\norm{\mathbf a \times \mathbf b} = 0$.

Hence, equality holds.

If both $\mathbf a$ or $\mathbf b$ are non-zero vectors, we have: