Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors \(3\mathbf{i}-

Question

Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $3\mathbf{i}-\mathbf{j}$, $5\mathbf{i}+3\mathbf{j}+4\mathbf{k}$ and $2\mathbf{i}+\alpha \mathbf{j}+\sqrt{2}\mathbf{k}$ respectively, where $\alpha $ is a non-zero constant. Given that the acute angle between the $x$-axis and the line passing through $A$ and $C$ is ${{60}^{{}^\circ }}$, find the value of $\alpha $.[3] The line $l$ passes through $A$ and $B$. A point $Q$ lies on $l$ such that $AQ=3$ units. Find the two possible position vectors of $Q$.[4] The plane $\pi $ containing $A$ has equation $x+2z=3$. Find the position vector of the point $F$ on $\pi $ that is closest to $B$.[3] Hence find a vector equation of the line of reflection of $l$ in $\pi $.[3]

Accepted Answer

(a) $\alpha =-2$ (b) $\lambda =1:\overrightarrow{OQ}=\left( \begin{matrix} \begin{matrix} 4 \ 1 \ 2 \ \end{matrix} \ \end{matrix} \right)$, $\lambda =-1:\overrightarrow{OQ}=\left( \begin{matrix} \begin{matrix} 2 \ -3 \ -2 \ \end{matrix} \ \end{matrix} \right)$ (c) $\overrightarrow{OF}=\left( \begin{matrix} \begin{matrix} 3 \ 3 \ 0 \ \end{matrix} \ \end{matrix} \right)$ (d) $\mathbf{r}=\left( \begin{matrix} \begin{matrix} 3 \ -1 \ 0 \ \end{matrix} \ \end{matrix} \right)+t\left( \begin{matrix} \begin{matrix} -1 \ 2 \ -2 \ \end{matrix} \ \end{matrix} \right)$, $t\in \mathbb{R}$

2023 NYJC Promo Q9

2023 NYJC Promo Q9

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