2023 ASRJC Promo Q11

With respect to the origin \(O\), the position vectors of the points \(A\) and \(B\) are \(\mathbf{i}+2\mathbf{j}+\mathbf{k}\) and \(\mathbf{i}-4\mathbf{k}\) respectively. The equation of the planes \({{p}_{1}}\) and \({{p}_{2}}\) are given by \({{p}_{1}}:\mathbf{r}\cdot \left( \begin{matrix} 1 \\ 3 \\ 0 \\ \end{matrix} \right)=1\) and \({{p}_{2}}:\mathbf{r}\cdot \left( \begin{matrix} 0 \\ 7 \\ \alpha \\ \end{matrix} \right)=-4\alpha \), where \(\alpha \) is a real constant with \(\alpha >0\).

1. Find the position vector of the foot of perpendicular from \(A\) to \(p_1\).[3]
2. Verify that \(B\) lies on both \(p_1\) and \(p_2\). Show that the vector equation of the line of intersection, \(L\), of \(p_1\) and \(p_2\) is of the form[2]

\(\mathbf{r}=\left( \begin{matrix} 1 \\ 0 \\ -4 \\ \end{matrix} \right)+s\left( \begin{matrix} 3\alpha \\ -\alpha \\ 7 \\ \end{matrix} \right)\), \(s\in \mathbb{R}\).
[2]

3. 1. The point \(C(-1, 1, -4)\) is equidistant from both \(p_1\) and \(p_2\). Calculate the value of \(\alpha\) for this case. Hence find in scalar product form the equation of the plane \(p_3\) in which both the line \(L\) and the point \(C\) lie.[5]
   2. The plane \(p_4\) has equation \(y - z = 2\). Given that the angle between \(L\) and \(p_4\) is \(\frac{\pi}{6}\), find the value of \(\alpha\).[3]