Unit 13.2 Worked Example 2

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The line \({{l}_{2}}\) passes through \(A\left( 3,1,-2 \right)\) and is parallel to the direction vector \(\left( \begin{matrix}7 \\15 \\1 \\\end{matrix} \right)\).

\({{l}_{2}}\) intersects \({{l}_{1}}:\mathbf{r}=\left( \begin{matrix} -12 \\ -22 \\ 11 \\\end{matrix} \right)+\lambda \left( \begin{matrix}4 \\ 4 \\ -7 \\\end{matrix} \right);\lambda \in \mathbb{R}\) at \(C\).

Find

(i)

the coordinates of \(C\),

(ii)

the acute angle between \({{l}_{1}}\) and \({{l}_{2}}\),

(iii)

the vector line equation of mirror image of \({{l}_{2}}\) in \({{l}_{1}}\).