Do not use a graphing calculator in answering this question. The planes ${{p}_{1}}$ and \({{p}_{2}

Question

Do not use a graphing calculator in answering this question. The planes ${{p}_{1}}$ and ${{p}_{2}}$ have equations $2x-z=3$ and $x-3y=-3$ respectively. The point $A$ with position vector $\lambda \mathbf{i}+\mathbf{j}+\mu \mathbf{k}$, where $\lambda $ and $\mu $ are constants, is in both ${{p}_{1}}$ and ${{p}_{2}}$. Find the values of $\lambda $ and $\mu $.[2] The planes ${{p}_{1}}$ and ${{p}_{2}}$ intersect in a line $l$. Find a vector equation of $l$.[2] The point $B$ has position vector $\mathbf{i}+2\mathbf{j}+4\mathbf{k}$. Find the position vector of $N$, the foot of the perpendicular from $B$ to${{p}_{1}}$. Hence find the vector equation of the line of reflection of $AB$ in ${{p}_{1}}$.[5]

Accepted Answer

(i) $\lambda=0,\ \mu=-3$ (ii) $\mathbf r=\begin{pmatrix}0\1\-3\end{pmatrix}+t\begin{pmatrix}3\1\6\end{pmatrix},\ t\in\mathbb R$ (iv) $\overrightarrow{ON}=\begin{pmatrix}3\2\3\end{pmatrix}$, reflected line $\mathbf r=\begin{pmatrix}0\1\-3\end{pmatrix}+\phi\begin{pmatrix}5\1\5\end{pmatrix},\ \phi\in\mathbb R$

2015 PJC P1 Q12

2015 PJC P1 Q12

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