Given a vector , write down the vector obtained by rotating through an angle .
Given a unit vector , any vector may be written as where is parallel to and is perpendicular to . Write down explicit formulae for and , in terms of and . Hence, or otherwise, show that the linear map
describes a rotation about through an angle , in the positive sense defined by the right hand rule.
Write equation in matrix form, . Show that the trace .
Given the rotation matrix
where , find the two pairs , with , giving rise to . Explain why both represent the same rotation.