Question

Show that the points with position vectors $\overrightarrow{a}+\overrightarrow{b},\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{a}+k\overrightarrow{b}$ are collinear for all values of $k$.

Answer 1

If the three points are $P,Q,R$ respectively, then $\overrightarrow{PQ}\times \overrightarrow{PR}=(\overrightarrow{OQ}-\overrightarrow{OP})\times (\overrightarrow{OR}-\overrightarrow{OP})$
$=(\overrightarrow{a}-\overrightarrow{b}-\overrightarrow{a}-\overrightarrow{b})\times (\overrightarrow{a}+\hat{k}\overrightarrow{b}-\overrightarrow{a}-\overrightarrow{b})$
$=(-2\overrightarrow{b})\times \left((k-1)\overrightarrow{b}\right)$
$=0$ since $\overrightarrow{b}\times \overrightarrow{b}=0$