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, where $\left| \mathbf{a} \right|=2$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{3}{4}\pi $ radians. Given that $\mathbf{a}$ and $2\mathbf{a}+\mathbf{b}$ are perpendicular, find the exact length of $\mathbf{b}$,[3] the exact length of projection of $\mathbf{a}$ onto $\mathbf{b}$.[1] The point $P$ lies on $OB$ such that the ratio $OP:OB=3:5$ . Find the exact area of triangle $APB$.[4]

Accepted Answer

(i) $\left| \mathbf{b} \right|=4\sqrt{2}$ (ii) $\sqrt{2}$ (iii) $\frac{8}{5}$

2015 PJC P1 Q5

2015 PJC P1 Q5

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