2023 MI PRU1 P1 Q5

With reference to the origin \(O\), the position vectors of points \(A\), \(B\) and \(C\) are expressed as \(\overrightarrow{OA}\), \(\overrightarrow{OB}\) and \(\overrightarrow{OC}\) respectively such that

\(\overrightarrow{OA}=\left( 1-\lambda \right)\overrightarrow{OB}+\lambda \overrightarrow{OC}\),

where \(\lambda \) is a non-zero constant.

1. Show that \(A\), \(B\), \(C\) are collinear.[2]

It is given that \(\lambda =\frac{1}{6}\), \(\overrightarrow{OA}=\left( \begin{matrix} \begin{matrix} 0 \\ -2 \\ 0 \\ \end{matrix} \\ \end{matrix} \right)\) and \(\overrightarrow{OB}=\left( \begin{matrix} \begin{matrix} 1 \\ 1 \\ -2 \\ \end{matrix} \\ \end{matrix} \right)\).

2. Find the position vector of the point \(D\) that lies on the line segment \(AC\) such that \(AD:DC=1:4\).[3]

It is given that the point \(E\) has coordinates \(\left( 1,-1,2 \right)\).

3. Find the exact area of the triangle \(ABE\).[3]
4. Find the length of projection of \(\overrightarrow{OE}\) onto \(\overrightarrow{AB}\).[2]