2008 NJC P1 Q5
The points $A$ and $B$ relative to the origin $O$ have position vectors $\mathbf{i}+2\mathbf{j}-2\mathbf{k}$ and $-4\mathbf{i}+5\mathbf{j}+2\mathbf{k}$respectively. The point P lies on line AB such that $\frac{AP}{PB}=\frac{\lambda }{1-\lambda }$.
(i)
Show that $\overrightarrow{OP}=(1-5\lambda )\mathbf{i}+(2+3\lambda )\mathbf{j}+(4\lambda -2)\mathbf{k}$.
(ii)
Given further that $C$ is a point with a position vector $-5\mathbf{i}+16\mathbf{j}-2\mathbf{k}$ and that $\overrightarrow{OP}=\mu \overrightarrow{OC}$, show that $\mu =\frac{1}{5}$and find the value of $\lambda$.
(iii)
Hence, or otherwise, determine the area of a triangle $OAP$, leaving your answer correct to $3$ decimal places.
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