The points $P$ and $Q$ have position vectors $p$ and $q$ respectively, which are non-zero and non-parallel. The points $P$ and $Q$ are fixed and $R$, with position vector $r$, varies.

(a) Given that $(2 r – p ) \times p = 0$, describe geometrically the set of all possible position vectors of $r$.

[2]

(b) If $r = q +\lambda p$, where $\lambda<0$, show that the area of $\triangle P Q R$ is $k| p \times q |$ where $k$ is a constant to be determined in terms of $\lambda$.

[3]

(a) $r$ represents the set of position vectors of points that lies on the line through the origin and parallel to $p$.

(b) $k=-\lambda / 2$

(a) $(2 r – p ) \times p = 0$ means $2 r – p$ is parallel to $p$ $\implies $ $r – p =k p$

$\implies r =(k+1) p$

$\implies r =\frac{k+1}{2} p$

$r$ represents the set of position vectors of points that lies on the line through the origin and parallel to $p$.

(b) Area $\triangle P Q R=\frac{1}{2}|\overrightarrow{Q R} \times \overrightarrow{Q P}|$

$=\frac{1}{2}|\lambda p \times(p-q)|$

$=\frac{1}{2}|\lambda(p \times q)|$

$=\frac{-\lambda}{2}|(p \times q)|$

$k=-\lambda / 2$