Thread: Composition or two reflections

Hi,
In Euclidean space I know that the composition of two reflections is a rotation if the two lines you wish to reflect in cross. I need to prove this. It is for credit so NO ANSWERS PLEASE, but some hints at how to get started would be very much appreciated. If I fix a choice of coordinates in R^2 does that mean I can assume the two lines cross at the origin, without loss of generality, and then pick a frame? Or by picking a specific frame am I only proving it for that frame? Basically, I understand what is going on, I just dont know how to rigorously prove it and show that in this case you will always get a rotation of 2*theta about the point where the two lines of reflection cross, where theta is the angle between the two lines.
Thanks,
Sooz

Originally Posted by Sooz

Hi,
In Euclidean space I know that the composition of two reflections is a rotation if the two lines you wish to reflect in cross. I need to prove this. It is for credit so NO ANSWERS PLEASE, but some hints at how to get started would be very much appreciated. If I fix a choice of coordinates in R^2 does that mean I can assume the two lines cross at the origin, without loss of generality, and then pick a frame? Or by picking a specific frame am I only proving it for that frame? Basically, I understand what is going on, I just dont know how to rigorously prove it and show that in this case you will always get a rotation of 2*theta about the point where the two lines of reflection cross, where theta is the angle between the two lines.
Thanks,
Sooz

Since the point where the lines cross is left unchanges by the two reflections if the reflections are equivalent to a rotation it is about the crossing point.

Now choose one line as one of the axes and the crossing point as the origin, and take a general point and work out its image under the two reflections, and show that this can be represented as a rotation independent of which point was choosen.

CB