The line $l$ passes through the point $A$ with coordinates $\left( 1,\,3,\,4 \right)$ and is p

Question

The line $l$ passes through the point $A$ with coordinates $\left( 1,\,3,\,4 \right)$ and is parallel to $2\mathbf{j}+\mathbf{k}$. State the vector equation of line $l$.[1] The plane $p$ has equation $r\cdot \left( \begin{matrix} 2 \ -1 \ -1 \ \end{matrix} \right)=4$. Find the acute angle between $l$ and $p$.[2] Find the position vector of the foot of perpendicular from $A$ to $p$.[3] $l$ and $p$ intersect at the point $X\left( 1,\,-3,\,1 \right)$. Using the result in (iii), find the vector equation of the line of reflection of $l$ in $p$.[3]

Accepted Answer

(i) $l:\mathbf r=\begin{pmatrix}1\3\4\end{pmatrix}+\lambda\begin{pmatrix}0\2\1\end{pmatrix}$, $\lambda\in\mathbb R$ (ii) $33.2^\circ$ (iii) $\overrightarrow{OF}=\begin{pmatrix}4\ \frac32\ \frac52\end{pmatrix}$ (iv) $l':\mathbf r=\begin{pmatrix}1\-3\1\end{pmatrix}+\mu\begin{pmatrix}2\1\0\end{pmatrix}$, $\mu\in\mathbb R$

2022 SAJC Promo Q5

2022 SAJC Promo Q5

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