Exercises 3

1. Find a condition for the product of two rotations in I(R²) to be a translation.
   Prove that the product of two reflections in lines in R² is a rotation if the lines meet and a translation otherwise.
   When is the product of two glide reflections a rotation ?

   Solution to question 1

2. Prove that the composite of a rotation about a point a and reflection R_L in a line L is a reflection if and only if a lies on the line L and a glide reflection otherwise.
   [Hint: Write the rotation as a product of reflection in a line parallel to L and another reflection.]

   Solution to question 2

3. Let H_a be a half-turn (rotation by π) about a point a in the plane. Prove that any translation can be written as a product of half-turns.

   Find and prove a condition for the composite H_a R_L to be a reflection.

   Prove that the composite H_a H_b H_c is a half-turn about the point a - b + c .

   Solution to question 3

4. Prove that the group I(R) of symmetries of a line is isomorphic to the group of matrices { | a, b ∈ R, b = ±1 } under the usual matrix multiplication.
   [Hint: Show that this group maps the set of points of the form to itself.]
   Generalise this to find a group of matrices isomorphic to I(R²).

   Solution to question 4

5. If AB is a (directed) line in R², let G_AB be the glide reflection along AB.
   Prove that "gliding around a rectangle" gives the identity.
   i.e. If ABCD is a rectangle, then G_DA G_CD G_BC G_AB = identity.
   Show that gliding around a general quadrilateral in the plane does not in general give the identity and find a condition that ensures that gliding around it does give the identity.

   Solution to question 5

6. If h, k are any symmetries, prove that hkh^-1k^-1 is a direct symmetry (a rotation or translation). If h and k are both direct, what can you say about hkh^-1k^-1 ?
   Show that any direct symmetry can be written as hkh^-1k^-1.

   Solution to question 6

7. Write down the 2 × 2 matrix which represents an anticlockwise rotation of 90° about the origin in R² (w.r.t. the standard basis). Hence write down 3 × 3 matrices representing rotations of 90° about the x and y axes in R³.
   Find the axis of the rotation got by combining these last two rotations.

   Solution to question 7