Exercises 2

1. Prove that an element of the orthogonal group O(3) acting on R³ is either:
   
   1. a rotation about some axis a,
      
   2. a reflection in some plane,
      
   3. a rotation about an axis a followed by a reflection in a plane perpendicular to a.
      
   
   Which of these three possibilities are the following ?
   
   1. Successive reflections in two planes meeting in a line b,
      
   2. The transformation x ↦ -x,
      
   3. Successive rotations by π/2 about three mutually perpendicular intersecting axes,
      
   4. Successive reflections in three mutually perpendicular planes.
      
   
   Solution to question 1

2. Show that any rotation acting on the plane R² can be written as a product of two reflections.
   Hence prove that any element of O(3) can be written as a product of at most three reflections in planes in R³.

   Solution to question 2

3. When we write an element of the group I(Rⁿ) of isometries of Rⁿ as a product T ∘ L with T a translation and L an orthogonal map, show that T and L are uniquely determined.
   Show that one can also write an isometry as L' ∘ T' with L' an orthogonal map and T' a translation.

   Solution to question 3

   

4. In the diagram on the right ABCD and XYZT are two equal squares. Find how many symmetries of R² map one of these squares into the other and describe all such symmetries.

   Solution to question 4
   
   
   

5. In the diagram on the right all the triangles are equilateral. Find how many symmetries of R² map the left-hand triangle into the right-hand one and describe all such symmetries.

   Solution to question 5
   
   
   

6. Let ABC and A'B'C' be two congruent plane triangles.
   If ABC and A'B'C' both go clockwise then prove that the perpendicular bisectors of AA', BB' and CC' either meet at a point or are parallel.
   If ABC goes clockwise and A'B'C' goes anticlockwise then prove that the midpoints of AA', BB' and CC' lie on a line.
   [Hint: look at an isometry taking ABC to A'B'C'.]

   Solution to question 6