$OACB$ is a square with origin $O$. Points $A$ and $B$ are such that \(\overrightarrow{

Question

$OACB$ is a square with origin $O$. Points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$. $M$ and $N$ are midpoints of $OB$ and $AC$ respectively. $S$ is on $AC$ such that $AS:AC=1:4$. $MS$ produced and $BN$ produced meet at point $T$. Find the position vector $\overrightarrow{OS}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.[2] Find vector equations of the line $MS$ and the line $BN$.[3] By finding the position vector of $T$, deduce that $T$, $A$ and $O$ are collinear points.[2]

Accepted Answer

(i) $\overrightarrow{OS}=\mathbf a+\dfrac14\mathbf b$ (ii) $MS:\ \mathbf r=\dfrac12\mathbf b+\lambda\left(\mathbf a-\dfrac14\mathbf b\right)$, $\lambda\in\mathbb R$, and $BN:\ \mathbf r=\mathbf b+\mu\left(\mathbf a-\dfrac12\mathbf b\right)$, $\mu\in\mathbb R$ (iii) $\overrightarrow{OT}=2\mathbf a$, so $O,A,T$ are collinear

2023 RVHS P1 Q5

2023 RVHS P1 Q5

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