Vectors: Reflection of a line in a line

I was talking with a student today about the reflection of a line in a line, where both lines are in the same plane (naturally). And I thought of sharing it. One reason why I decided to discuss this in class was also because A’levels has been moving towards questions that want students to engineer their own solutions and thought process from start to end. So here it is a question. 🙂 Students can try first before clicking solution.

Given l_1: \textbf{r} = \textbf{a} + \lambda \textbf{b} , \lambda \in \mathbb{R} and l_2: \textbf{r} = \textbf{a} + \mu \textbf{c} , \mu \in \mathbb{R}. Find, in terms of \textbf{a}, \textbf{b} and \textbf{c} , the vector equation of the line l_3, which is the reflection of l_1 about l_2, in the same plane.

Students might want to try to ask themselves, if the lines do not intersection \textbf{a} instead, how does that change the approach? Assuming, they will still intersect somewhere.

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