Like Unit Vectors are Equal

Theorem
Let $\mathbf a$ and $\mathbf b$ be like vector quantities.

Then:
 * $\mathbf {\hat a} = \mathbf {\hat b}$

where $\mathbf {\hat a}$ and $\mathbf {\hat b}$ denote the unit vectors in the direction of $\mathbf a$ and $\mathbf b$.

Proof
By definition of like vector quantities, $\mathbf a$ and $\mathbf b$ have the same direction.

By definition of unit vector, $\mathbf {\hat a}$ and $\mathbf {\hat b}$ are both of magnitude $1$.

Hence the result, by Equality of Vector Quantities.