Equality of Vector Quantities

Theorem
Two vector quantities are equal they have the same magnitude and direction.

That is:
 * $\mathbf a = \mathbf b \iff \paren {\size {\mathbf a} = \size {\mathbf b} \land \hat {\mathbf a} = \hat {\mathbf b} }$

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

Proof
Let $\mathbf a$ and $\mathbf a$ be expressed