2013 RVHS P1 Q9
Referred to an origin, $O$, the position vectors of two points $A$ and $B$ are $\mathbf{a}$ and $\mathbf{b}$ respectively. $\mathbf{a}$ and $\mathbf{b}$ are not parallel.
(i)
The point $C$ lies on $AB$ produced such that $AB:BC$ is $1:3$. Find the position vector of $C$.
[2]
(ii)
The point $D$ lies on $OB$ produced such that $OB:OD$ is $1:k$. Given also that $CD$ is perpendicular to $AB$, show that $k=\frac{4{{\left| \mathbf{b} \right|}^{2}}-7\mathbf{a}\cdot \mathbf{b}+3{{\left| \mathbf{a} \right|}^{2}}}{{{\left| \mathbf{b} \right|}^{2}}-\mathbf{a}\cdot \mathbf{b}}$.
[3]
(iii)
Show that $OADC$ cannot be a parallelogram.
[2]
(iv)
Find the area of triangle $ABD$ in terms of $k$.
[3]
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