Relative to the origin $O$, the position vectors of the points $A$ and $B$ are given by \(\mat

Question

Relative to the origin $O$, the position vectors of the points $A$ and $B$ are given by $\mathbf{a}$ and $\mathbf{b}$ respectively. The point $P$ is on $OA$ such that $OP = 3PA$ and point $Q$ is on $AB$ such that $AQ = kAB$, where $k$ is a constant and $0 < k < 1$. Express $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $k$, where applicable.[2] Point $R$ is on $PQ$ produced such that $PQ : QR = 4 : 5$. Given that the points $O$, $B$ and $R$ are collinear, find the value of $k$.[3] Show that the area of triangle $OPR$ can be written as $s\left| \mathbf{a}\times \mathbf{b} \right|$, where $s$ is a constant to be found.[2]

Accepted Answer

(i) ${\overrightarrow{OQ}} = (1 - k)\mathbf{a} + k\mathbf{b}$ (ii) ${k = \frac{7}{12}}$ (iii) ${s = \frac{63}{128}}$

2024 MI PRU2 P2 Q3

2024 MI PRU2 P2 Q3