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 such that $|\mathbf{a}|=5$ and $|\mathbf{b}|=3$. It is given that $|2\mathbf{b} - \mathbf{a}| = 7$. Using a suitable scalar product, show that $\mathbf{a} \cdot \mathbf{b} = 3$.[2] Deduce the length of projection of $\overrightarrow{OA}$ onto $\overrightarrow{OB}$.[1] The point $P$ lies on $AB$ such that $AP:PB = 1:k$, where $k$ is a positive constant. Write down the position vector of $P$ in terms of $k$, $\mathbf{a}$ and $\mathbf{b}$ and show that the area of triangle $OAP$ is $\frac{3\sqrt{6}}{k+1}$.[5]

Accepted Answer

(a) $\mathbf a\cdot\mathbf b=3$ (b) $1$ (c) $\overrightarrow{OP}=\frac{k\mathbf a+\mathbf b}{1+k}$, area $=\frac{3\sqrt6}{1+k}$

Vectors - Projection Lengths

Vectors - Projection Lengths

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