2022 SAJC J2 MYE Q5
(a)
Relative to the origin $O$, the fixed points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$. The point $R$ has position vector $\overrightarrow{OR}=\left( \begin{matrix}
x \\
y \\
z \\
\end{matrix} \right)$. Interpret geometrically the point $R$, stating what $d$ represents, given that $\overrightarrow{OR}\cdot \frac{\left( \mathbf{a}\times \mathbf{b} \right)}{\left| \mathbf{a}\times \mathbf{b} \right|}=d$ where $d$ is a constant scalar.
[2]
(b)
The points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively relative to the origin $O$. $M$ is the mid- point of $AB$. The point $N$ on $OB$ is such that $ON:NB=k:1-k$, where $0<k<1$.
(i) The lines $OM$ and $AN$ intersect at the point $P$. Given that $OP:PM=2k:1-k$, show that $\overrightarrow{OP}=\frac{k}{1+k}\left( \mathbf{a}+\mathbf{b} \right)$.
[2]
(ii) It is given that $\left| \mathbf{a}\cdot \mathbf{b} \right|=3$ and $\mathbf{a}$ is a unit vector. Find $\left| \mathbf{b} \right|$ if the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{\pi }{6}$.
Hence, find the area of triangle $OPN$ in terms of $k$.
[4]
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