2021 ASRJC Promo Q12

Relative to the origin \(O\), a point \(A\) has position vector \(-\mathbf{j}+2\mathbf{k}\). The plane \({{p}_{1}}\) has equation \(\mathbf{r}\cdot \left( \begin{matrix} 1 \\-1 \\0 \end{matrix} \right)=3\).

1. Find the position vector of the foot of perpendicular from point \(A\) to the plane \({{p}_{1}}\).[4]

The line \(l\) has equation \(\mathbf{r}=\left( \begin{matrix} 0 \\-1 \\2 \end{matrix} \right)+\mu \left( \begin{matrix}0 \\1 \\1 \\ \end{matrix} \right)\),\(\mu \in \mathbb{R}\).

2. Find the acute angle between the plane \({{p}_{1}}\) and the line \(l\).[2]
3. The point \(B\left( -\alpha ,2,\alpha \right)\) is equidistant from the plane \({{p}_{1}}\) and the line \(l\). Find the possible values of \(\alpha \).[4]

The plane \({{p}_{2}}\) has equation \(\mathbf{r}\cdot \left( \begin{matrix} 1 \\1 \\-1 \end{matrix} \right)=5-\beta \), \(\beta \ne 8\).

4. Show that the point \(C\left( 4,1,\beta \right)\) lies on both \({{p}_{1}}\) and \({{p}_{2}}\). Hence find the vector equation of the line of intersection between the planes \({{p}_{1}}\) and \({{p}_{2}}\)[3]