Referred to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and \(

Question

Referred to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-parallel vectors. The lines $AB$ and $OC$ intersect at point $X$ such that $AX:XB=3:1$ and $OX:XC=3:5$(see diagram). Express $\overrightarrow{OC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.[2] Given that $OC$ is perpendicular to $AB$ and $OA:OB=3:2$, show that $\mathbf{a}\cdot \mathbf{b}=\frac{3}{8}{{\left| \mathbf{b} \right|}^{2}}$.[3] Hence, by considering ${{\left| \overrightarrow{OC} \right|}^{2}}$, find $OC:OB$.[3]

Accepted Answer

(a) $\overrightarrow{OC}=\dfrac23(\mathbf a+3\mathbf b)$ (b) $\mathbf a\cdot\mathbf b=\dfrac38|\mathbf b|^2$ (c) $OC:OB=\sqrt6:1$

2022 EJC Promo Q7

2022 EJC Promo Q7

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