# Vector Quantity as Scalar Product of Unit Vector Quantity

## Theorem

Let $\mathbf a$ be a vector quantity.

Then:

$\mathbf a = \size {\mathbf a} \mathbf {\hat a}$

where:

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

## Proof

 $\ds \size {\mathbf {\hat a} }$ $=$ $\ds 1$ Definition of Unit Vector $\ds \leadsto \ \$ $\ds \mathbf {\hat a} \times \size {\mathbf {\hat a} }$ $=$ $\ds \mathbf {\hat a}$ $\ds \leadsto \ \$ $\ds \size {\mathbf a} \times \size {\mathbf {\hat a} } \times \mathbf {\hat a}$ $=$ $\ds \size {\mathbf a} \times \mathbf {\hat a}$ $\ds \leadsto \ \$ $\ds \mathbf a$ $=$ $\ds \size {\mathbf a} \mathbf {\hat a}$

$\blacksquare$