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

 * Scalar-product-distributes-over-vector-addition.png

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: