2023 TJC Promo Q10

The line \(l\) has equation \(\mathbf{r}=\left( \begin{matrix} \begin{matrix} 2 \\ -1 \\ 0 \\ \end{matrix} \\ \end{matrix} \right)+\lambda \left( \begin{matrix} \begin{matrix} 1 \\ 0 \\ 3 \\ \end{matrix} \\ \end{matrix} \right)\) where \(\lambda \) is a parameter. The point \(A\) has coordinates \(\left( 2,-2,1 \right)\).

1. Find the position vector of the foot of perpendicular from \(A\) to \(l\).[3]

The plane \({{p}_{1}}\) contains \(A\) and \(l\).

2. Find, in scalar product form, a vector equation of \({{p}_{1}}\).[3]

It is given that \(l\) also lies on a second plane \({{p}_{2}}\) with equation \(\mathbf{r}\cdot \left( \begin{matrix} \begin{matrix} 1 \\ a \\ b \\ \end{matrix} \\ \end{matrix} \right)=-1\) where \(a\) and \(b\) are constants.

3. Show that \(b=-\frac{1}{3}\) and find the value of \(a\).[3]
4. Hence find the acute angle between \({{p}_{1}}\) and \({{p}_{2}}\).[2]