NYJC/2021/JC1/Promo/P1/Q03

Referred to the origin $O$, points $A$ and $B$ have position vectors a and $\mathbf{b}$ respectively. Point $P$ lies on $O A$ such that $O P=2 P A$ and point $Q$ lies on $A B$ such that $5 A Q=4 Q B$. Show that the equation of the line $l$ passing through $P$ and $Q$ can be written as
$$\mathbf{r}=\frac{2}{3} \mathbf{a}+\lambda(4 \mathbf{b}-\mathbf{a}) \text {, where } \lambda \in \mathbb{R} .$$

[3]

Point $X$ lies on $/$ such that $A X$ is perpendicular to $l$. If $|\mathbf{a}|=\sqrt{3},|\mathbf{b}|=\frac{1}{2}$ and $\mathbf{a}$ is perpendicular to $\mathbf{b}$, find the position vector of $X$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

[4]