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 and . Find, in terms of and , the vector equation of the line , which is the reflection of about , in the same plane.
Observe that is a common point of intersection between and . Thus, it should also lie on . We also consider and have that is a point, on and be the point of reflection of in . Let the foot of perpendicular of to be . Then for some .
Students might want to try to ask themselves, if the lines do not intersection instead, how does that change the approach? Assuming, they will still intersect somewhere.