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 \(\frac{AP}{PB}=\frac{\lambda }{1-\lambda }\).

1. Show that \(\overrightarrow{OP}=(1-5\lambda )\mathbf{i}+(2+3\lambda )\mathbf{j}+(4\lambda -2)\mathbf{k}\).
2. 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 \(\lambda\).
3. Hence, or otherwise, determine the area of a triangle \(OAP\), leaving your answer correct to \(3\) decimal places.