Referred to the origin $O$, the points $P$ and $Q$ have position vectors $\mathbf{p}$ and \(

Question

Referred to the origin $O$, the points $P$ and $Q$ have position vectors $\mathbf{p}$ and $\mathbf{q}$ respectively, where vector $\mathbf{p}$ and $\mathbf{q}$ are not parallel. Point $A$ lies between $P$ and $Q$, such that $PA:AQ=\lambda :\left( 1-\lambda \right)$, where $0<\lambda <1$. Express $\overrightarrow{OA}$ in terms of $\mathbf{p}$, $\mathbf{q}$ and $\lambda $.[1] The point $B$ lies between $O$ and $Q$, such that $OB:OQ=1:3$. Given that $\overrightarrow{AB}$ is parallel to $\left( \mathbf{q}-4\mathbf{p} \right)$, find the value of $\lambda $.[4] It is now given that $\mathbf{p}$ is a unit vector which is perpendicular to $\mathbf{q}$ and $\left| \mathbf{q} \right|=2$, find the area of triangle $OAQ$, in terms of $\lambda $.[4]

Accepted Answer

(i) $\overrightarrow{OA}=(1-\lambda)\mathbf p+\lambda\mathbf q$ (ii) $\lambda=\dfrac19$ (iii) area of $\triangle OAQ=1-\lambda\text{ units}^2$

2023 ACJC Promo Q6

2023 ACJC Promo Q6

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