The line ${{L}_{1}}$ has equation $1-y=\frac{z-1}{2}$, $x=2$, and meets the $xy$-plane at po

Question

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 $O$. Find a vector equation of the line ${{L}_{2}}$ which passes through $O$ and $P$.[3] Find an equation of the plane $\pi $ containing both ${{L}_{1}}$ and ${{L}_{2}}$, in the scalar product form.[2] 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] 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]

Accepted Answer

(i) ${L_2 : \mathbf{r} = \mu \begin{pmatrix} 4 \ 3 \ 0 \end{pmatrix} ; \mu \in \mathbb{R}}$ (ii) ${\pi : \mathbf{r} \cdot \begin{pmatrix} -3 \ 4 \ 2 \end{pmatrix} = 0}$ (iii) ${\overrightarrow{OC} = \frac{1}{29} \begin{pmatrix} 60 - 27t \ 7 + 36t \ 76 + 18t \end{pmatrix}}$ (iv) ${t = \frac{49}{9}}$

2022 RI P2 Q3

2022 RI P2 Q3

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