Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 8B / b / ii

Table of contents
- Solution ii

Solution

Choose the

z

-axis along the line of intersection of the two planes. Their unit normals lie in the

x y

-plane; after choosing the

x

-axis, take

n = (1, 0, 0), m = (cos θ, sin θ, 0),

where

θ

is the oriented angle between the planes. The two reflection matrices restrict to

H_{n} = (- 1 0 01), H_{m} = (- cos 2 θ - sin 2 θ - sin 2 θ cos 2 θ)

on the

x y

-plane and both fix the

z

-axis. Their product is

H_{m} H_{n} = (cos 2 θ sin 2 θ - sin 2 θ cos 2 θ) .

Thus the composition of two plane reflections is a rotation about the intersection line through angle $2 θ$ .

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