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 length of $\mathbf{a}$ is $2$ units and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{\pi }{6}$ radians. It is also given that the vector $\mathbf{a}$ and $\mathbf{b}-2\mathbf{a}$ are perpendicular. Find the exact length of projection of $\mathbf{a}$ onto $\mathbf{b}$,[2] the exact length of $\mathbf{b}$.[3] The point $P$ lies on $OB$ such that the ratio $OP:OB=3:4$. Find the exact area of triangle $APB$.[4]

Accepted Answer

(i) $\sqrt3\text{ units}$. (ii) $\frac8{\sqrt3}\text{ units}$. (iii) $\frac1{\sqrt3}\text{ units}^2$.

2014 NYJC P1 Q5

2014 NYJC P1 Q5

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