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- Prove that an element of the orthogonal group
*O*(3) acting on**R**^{3}is either:

- a rotation about some axis
*a*,

- a reflection in some plane,

- a rotation about an axis
*a*followed by a reflection in a plane perpendicular to*a*.

Which of these three possibilities are the following ?

- Successive reflections in two planes meeting in a line
*b*,

- The transformation
↦ -*x*,*x*

- Successive rotations by π/2 about three mutually perpendicular intersecting axes,

- Successive reflections in three mutually perpendicular planes.

Solution to question 1 - a rotation about some axis
- Show that any rotation acting on the plane
**R**^{2}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**R**^{3}. - When we write an element of the group
*I*(**R**^{n}) of isometries of**R**^{n}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. - In the diagram on the right
*ABCD*and*XYZT*are two equal squares. Find how many symmetries of**R**^{2}map one of these squares into the other and describe all such symmetries. - In the diagram on the right all the triangles are equilateral. Find how many symmetries of
**R**^{2}map the left-hand triangle into the right-hand one and describe all such symmetries. - 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*'.]

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