Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{i}+2\mathbf{j

Question

Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{i}+2\mathbf{j}+\mathbf{k}$, $2\mathbf{i}-\mathbf{j}+3\mathbf{k}$ and $4\mathbf{i}-7\mathbf{j}+7\mathbf{k}$ respectively. Show that $A$, $B$ and $C$ are collinear. [2] The line ${{l}_{1}}$ passes through the points $A$ and $B$. The line ${{l}_{2}}$ has equation $\mathbf{r}=\left( \begin{matrix}1 \2 \1 \\end{matrix} ight)+\mu \left( \begin{matrix}1 \0 \-1 \\end{matrix} ight)$, $\mu \in \mathbb{R}$ and passes through the points $A$ and $D$, where $CD$ is perpendicular to line ${{l}_{1}}$. Show that $D$ has coordinates $\left( -41,\,\,2,\,\,43 ight)$. [4] Find a vector equation of the line ${{l}_{3}}$ which is a reflection of the line ${{l}_{2}}$ in the line ${{l}_{1}}$. [3] Hence or otherwise, find the acute angle between lines ${{l}_{2}}$ and ${{l}_{3}}$. [2]

Accepted Answer

(i) $\overrightarrow{AC} = 3\overrightarrow{AB}$ and $A$ is the common point, so $A$, $B$, $C$ are collinear. (ii) $(-41, 2, 43)$ (iii) $l_3: \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} + d\begin{pmatrix} 8 \\ -3 \\ -5 \end{pmatrix}, d \in \mathbb{R}$ (iv) $21.8°$

2017 CJC Promo Q10

2017 CJC Promo Q10

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