ACJC Tutorial 15 Q8

Given that a line \(l\) has equation \(\mathbf{r}=3\mathbf{i}-4\mathbf{j}+7\mathbf{k}+\lambda (-\mathbf{i}+3\mathbf{j}-2\mathbf{k})\), \(\lambda \in \mathbb{R}\).

1. Find the position vector of the point on line \(l\) closest to the point \(P(0,1,2)\). Hence show that the shortest distance from \(P\) to line \(l\) is \(\sqrt{3}\).
2. Find the position vector of the mirror image of \(P\) when \(P\) is reflected in the line \(l\).
3. The line \(m\) is parallel to the vector \(3\mathbf{i}-5\mathbf{j}+5\mathbf{k}\) and passes through the point \(P\). Find the equation of the line obtained when line \(m\) is reflected in the line \(l\).