2018 RVHS P1 Q4

Relative to the origin \(O\), the points \(A\), \(B\) and \(C\) are such that \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\) and \(\overrightarrow{OC}=\mathbf{c}\) respectively, where \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are vectors which are mutually non-parallel.

The plane \(\prod \) and the line \(l\) have the following equations

\(\prod :\mathbf{r}=\lambda \mathbf{a}+\mu \mathbf{b}\) where \(\lambda \), \(\mu \in \mathbb{R}\) and

\(l:\mathbf{r}=\mathbf{a}+\beta \mathbf{c}\) where \(\beta \in \mathbb{R}\)

respectively.

1. Find the point(s) of intersection between \(\prod \) and \(l\) given that the points \(O\), \(A\), \(B\) and \(C\) are
   1. coplanar,
   2. not coplanar.[3]
2. State the geometrical meaning of \(\frac{1}{2}\left| \mathbf{a}\times \mathbf{b} \right|\).[1]
   In the rest of the question, \(O\), \(A\), \(B\) and \(C\) are not coplanar.
3. The vector \(\mathbf{p}\) is a unit vector in the direction of \(\mathbf{a}\times \mathbf{b}\). State the geometrical meaning of \(\left| \mathbf{c}\cdot \mathbf{p} \right|\).[1]
4. It is given that \(\overrightarrow{OA}=\left( \begin{matrix}1 \\1 \\0 \end{matrix} \right)\), \(\overrightarrow{OB}=\left( \begin{matrix}0 \\1 \\1 \end{matrix} \right)\) and \(\overrightarrow{OC}=\left( \begin{matrix}0 \\0 \\2 \end{matrix} \right)\). Find the volume of the pyramid \(OABC\).[3]