Relative to the origin $O$, point $A$ has position vector given by \(\mathbf{a} = \begin{pmatrix

Question

Relative to the origin $O$, point $A$ has position vector given by $\mathbf{a} = \begin{pmatrix} 1 \ 2 \ 1 \end{pmatrix}$. The line $l_1$ passes through the point $A$ and is parallel to $\begin{pmatrix} t \ 3 - t^2 \ 1 \end{pmatrix}$. The plane $p$ has equation $\mathbf{r} = \lambda \begin{pmatrix} 0 \ 5 \ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \ 0 \ t \end{pmatrix}, \lambda, \mu \in \mathbb{R}$ where $t$ is a real constant. It is known that $l_1$ and $p$ are parallel and $l_1$ is not on $p$. Show that $t = -1$.[4] The line $l_2$ passes through the origin and point $A$. Find the acute angle between $l_1$ and $l_2$.[2] The vector $\mathbf{n}$ is a unit vector normal to $p$. State the geometrical meaning of $|\mathbf{a} \cdot \mathbf{n}|$ and find the exact value of $|\mathbf{a} \cdot \mathbf{n}|$.[3] Find a vector equation of the line of reflection of $l_2$ in $p$.[4]

Accepted Answer

(a) $t=-1$ (b) $48.2^\circ$ or $0.841\text{ rad}$ (c) Perpendicular distance from $A$ to $p$, equivalently from $l_1$ to $p$, $=\dfrac2{\sqrt{30}}=\dfrac{\sqrt{30}}{15}$ (d) $\mathbf r=m\begin{pmatrix}13\34\5\end{pmatrix}$, $m\in\mathbb R$

2025 EJC P2 Q4

2025 EJC P2 Q4

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