Position Vector of Midpoint of Line

Theorem

Let $\mathbf a$ and $\mathbf b$ be the position vectors of points $A$ and $B$.

The position vector $\mathbf r$ of the midpoint of the line segment $AB$ is given by:

$\mathbf r = \dfrac {\mathbf a + \mathbf b} 2$

the position vector $\mathbf r$ of a point $R$ on $AB$ which divides $AB$ in the ratio $m : n$ is given by:

$\mathbf r = \dfrac {n \mathbf a + m \mathbf b} {m + n}$

In this case the ratio $m : n$ is $1 : 1$.

Hence when $\mathbf r$ is the position vector of the midpoint of $AB$:

$\ds \mathbf r$

$=$

$\ds \dfrac {1 \times \mathbf a + 1 \times \mathbf b} {1 + 1}$

$\ds $

$=$

$\ds \dfrac {\mathbf a + \mathbf b} 2$

$\blacksquare$