Like Vector Quantities are Multiples of Each Other

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

Then:
 * $\mathbf a = \dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$

where:
 * $\size {\mathbf a}$ denotes the magnitude of $\mathbf a$
 * $\dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$ denotes the scalar product of $\mathbf b$ by $\dfrac {\size {\mathbf a} } {\size {\mathbf b} }$.

Proof
By the definition of like vector quantities:


 * $\mathbf a$ and $\mathbf b$ are like vector quantities they have the same direction.

By definition of unit vector:
 * $\dfrac {\mathbf a} {\size {\mathbf a} } = \dfrac {\mathbf b} {\size {\mathbf b} }$

as both are in the same direction, and both have length $1$.

By definition of scalar division:
 * $\dfrac 1 {\size {\mathbf a} } \mathbf a = \dfrac 1 {\size {\mathbf b} } \mathbf b$

Hence, multiplying by $\size {\mathbf a}$:
 * $\mathbf a = \dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$