2022 RI P2 Q3
The line ${{L}_{1}}$ has equation $1-y=\frac{z-1}{2}$, $x=2$, and meets the $xy$-plane at point $P$. The point $A$ has position vector $\left( \begin{matrix}
3 \\
-1 \\
2 \\
\end{matrix} \right)$ with reference to the origin $O$.
(i)
Find a vector equation of the line ${{L}_{2}}$ which passes through $O$ and $P$.
[3]
(ii)
Find an equation of the plane $\pi $ containing both ${{L}_{1}}$ and ${{L}_{2}}$, in the scalar product form.
[2]
(iii)
The points $A$ and $C$ are on different sides of $\pi $ such that $AC$ is perpendicular to $\pi $. The distance of $C$ from $\pi $ is $t$ times the distance of $A$ from $\pi $. Find, in terms of $t$, the position vector of $C$.
[5]
(iv)
Find the value of $t$ such that the line $OC$ is parallel to the plane with equation $\mathbf{r}\cdot \left( \begin{matrix}
2 \\
0 \\
1 \\
\end{matrix} \right)=2$.
[2]
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