# Scalar Multiplication of Vectors is Distributive over Vector Addition

## Theorem

Let $\mathbf a, \mathbf b$ be a vector quantities.

Let $m$ be a scalar quantity.

Then:

$m \paren {\mathbf a + \mathbf b} = m \mathbf a + m \mathbf b$

## Proof

Let $\mathbf a = \vec {OP}$ and $\mathbf b = \vec {PQ}$.

Then:

$\vec {OQ} = \mathbf a + \mathbf b$

Let $P'$ and $Q'$ be points on $OP$ and $OQ$ respectively so that:

$OP' : OP = OQ' : OQ = m$

Then $P'Q'$ is parallel to $PQ$ and $m$ times it in length.

Thus:

$\vec {P'Q'} = m \mathbf b$

which shows that:

 $\ds m \paren {\mathbf a + \mathbf b}$ $=$ $\ds \vec {OQ'}$ $\ds$ $=$ $\ds \vec {OP} + \vec {OP'}$ $\ds$ $=$ $\ds m \mathbf a + m \mathbf b$

$\blacksquare$