With respect to the origin $O$, the position vectors of two points $A$ and $B$ are given by $\mathbf

Question

With respect to the origin $O$, the position vectors of two points $A$ and $B$ are given by $\mathbf{a}$ and $\mathbf{b}$ respectively where $\left| \mathbf{a} \right|=1$,$\left| \mathbf{b} \right|=2$ and $\mathbf{a}\cdot \mathbf{b}=1$. Point $C$ lies on $AB$such that $AC=2CB$. Point $N$ is the foot of perpendicular from $C$ to the line $OB$. Find the position vector of the point $C$ in terms of $\mathbf{a}$ and $\mathbf{b}$. [1] Show that the length of projection of $\overrightarrow{OC}$ onto $\overrightarrow{OB}$ is $\frac{3}{2}$ and deduce the position vector of $N$. [4] Find the value of $\lambda $ such that $C$, $D$ and $N$ are collinear where $\overrightarrow{OD}=\mathbf{a}+\lambda \mathbf{b}$. [3]

Accepted Answer

(i) $\overrightarrow{OC} = \displaystyle\frac{\mathbf{a} + 2\mathbf{b}}{3}$; (ii) LOP $= \displaystyle\frac{3}{2}$ units, $\overrightarrow{ON} = \displaystyle\frac{3}{4}\mathbf{b}$; (iii) $\lambda = \displaystyle\frac{1}{2}$

2019 DHS Promo Q7

2019 DHS Promo Q7

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