Scalar Product of Magnitude by Unit Vector Quantity

Theorem
Let $\mathbf a$ be a vector quantity.

Let $m$ be a scalar quantity.

Then:
 * $m \mathbf a = m \paren {\size a \hat {\mathbf a} } = \paren {m \size a} \hat {\mathbf a}$

where:
 * $\size a$ denotes the magnitude of $\mathbf a$
 * $\hat {\mathbf a}$ denotes the unit vector in the direction $\mathbf a$.

Proof
Then: