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- Find a condition for the product of two rotations in
*I*(**R**^{2}) to be a translation.

Prove that the product of two reflections in lines in**R**^{2}is a rotation if the lines meet and a translation otherwise.

When is the product of two glide reflections a rotation ? - 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.] - 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**. - 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*(**R**^{2}). - If
*AB*is a (directed) line in**R**^{2}, 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. - If
*h*,*k*are any symmetries, prove that*hkh*^{-1}*k*^{-1}is a direct symmetry (a rotation or translation). If*h*and*k*are both direct, what can you say about*hkh*^{-1}*k*^{-1}?

Show that any direct symmetry can be written as*hkh*^{-1}*k*^{-1}. - Write down the 2 × 2 matrix which represents an anticlockwise rotation of 90° about the origin in
**R**^{2}(w.r.t. the standard basis). Hence write down 3 × 3 matrices representing rotations of 90° about the*x*and*y*axes in**R**^{3}.

Find the axis of the rotation got by combining these last two rotations.

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