Relative to the origin $O$, the points $A$, $B$, $C$ have position vectors $\mathbf{a}$, $\mathbf{b}

Question

Relative to the origin $O$, the points $A$, $B$, $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $3\mathbf{a}$ respectively. Point $P$ lies on $AB$ such that $AP:PB=1:2$ and point $Q$ lies on $OP$ produced that $5OQ=9OP$. Find the position vector of $Q$ in terms of $\mathbf{a}$ and $\mathbf{b}$. [2] Show that $Q$ lies on $BC$ and hence state the ratio $BQ:QC$. [3] Show that the area of triangle $APQ$ is $k\left| \mathbf{a}	imes \mathbf{b} ight|$, where $k$ is a constant to be determined. [2]

Accepted Answer

(i) OQ = (3/5)(b + 2a); (ii) BQ : QC = 2 : 3; (iii) k = 2/15

2014 PJC P1 Q3

2014 PJC P1 Q3

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