The line ${{l}_{1}}$ passes through the points $A$ and $B$ with coordinates $(-1,-2,\,\,1)$ and $(0,

Question

The line ${{l}_{1}}$ passes through the points $A$ and $B$ with coordinates $(-1,-2,\,\,1)$ and $(0,\,\,1,\,\,5)$ respectively. The line ${{l}_{2}}$ has equation $x-1=\frac{y-2}{2}=z-3$. ${{l}_{1}}$ and ${{l}_{2}}$ intersect at the point $A$. Find the vector equations of the lines ${{l}_{1}}$ and ${{l}_{2}}$, [2] the acute angle between the lines ${{l}_{1}}$ and ${{l}_{2}}$, [2] the position vector of the foot of perpendicular, $F$, from $B$ to the line ${{l}_{2}}$, [3] the equation of the line ${{l}_{3}}$ which is the mirror image of ${{l}_{1}}$ in ${{l}_{2}}$. [3]

Accepted Answer

(a)(i) $l_1: \mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 5 \end{pmatrix} + \lambda\begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $l_2: \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \mu\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$ (a)(ii) $28.3°$ (a)(iii) $\overrightarrow{OF} = \frac{1}{6}\begin{pmatrix} 5 \\ 10 \\ 17 \end{pmatrix}$ (a)(iv) $l_3: \mathbf{r} = \begin{pmatrix} -1 \\ -2 \\ 1 \end{pmatrix} + d\begin{pmatrix} 8 \\ 13 \\ -1 \end{pmatrix}$

2012 CJC Promo Q10

2012 CJC Promo Q10

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