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 $\mathbf{r}\cdot \left( \begin{matrix}
1 \\-1 \\0 \\\end{matrix} \right)=3$.
(i)
Find the position vector of the foot of perpendicular from point $A$ to the plane ${{p}_{1}}$.
[4]
The line $l$ has equation $\mathbf{r}=\left( \begin{matrix}
0 \\-1 \\2 \\\end{matrix} \right)+\mu \left( \begin{matrix}0 \\1 \\1 \\
\end{matrix} \right)$,$\mu \in \mathbb{R}$.
(ii)
Find the acute angle between the plane ${{p}_{1}}$ and the line $l$.
[2]
(iii)
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 $, $\beta \ne 8$.
(iv)
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]
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