Finding the Position Vector of a Point

The section formula

A point $R$ dividing the segment from $P$ to $Q$ in the ratio $m:n$ has position vector:

Worked example

Find the position vector of $R$, which divides the segment joining $P$ (position vector $\hat{i}+2\hat{j}-\hat{k}$) and $Q$ (position vector $-\hat{i}+\hat{j}+\hat{k}$) in the ratio 2:1, both internally and externally.

Solution, internally ($m=2, n=1$):

Externally:

Worked example: a midpoint

Find the position vector of the midpoint of the segment joining $P(2,3,4)$ and $Q(4,1,-2)$.

Solution: A midpoint is just the 1:1 case of the section formula: average each coordinate: