Relative to the origin $O$, the position vectors of two points $A$ and $B$ are $\mathbf{a}$

Question

Relative to the origin $O$, the position vectors of two points $A$ and $B$ are $\mathbf{a}$ and $\mathbf{b}$ respectively. The vector $\mathbf{a}$ is a unit vector which is perpendicular to $\mathbf{a}+3\mathbf{b}$ . The angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{2\pi }{3}$ . Show that $\left| \mathbf{b} \right|=\frac{2}{3}$ .[4] By expanding $\left( \mathbf{b}-2\mathbf{a} \right)\centerdot \left( \mathbf{b}-2\mathbf{a} \right)$, find the exact value of $\left| \mathbf{b}-2\mathbf{a} \right|$.[4] The point $P$ divides the line $AB$ in the ratio $\lambda:1-\lambda$ . Find the area of triangle $OAP$ in terms of $\lambda$.[5]

Accepted Answer

(ii) $\frac{2\sqrt{13}}{3}$ (iii) $\frac{\lambda \sqrt{3}}{6}$

2011 IJC P2 Q4

2011 IJC P2 Q4

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