Paper 1, Section II, B

(a) Consider the matrix

$R=\left(\begin{array}{ccc} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array}\right)$

representing a rotation about the $z$-axis through an angle $\theta$.

Show that $R$ has three eigenvalues in $\mathbb{C}$ each with modulus 1 , of which one is real and two are complex (in general), and give the relation of the real eigenvector and the two complex eigenvalues to the properties of the rotation.

Now consider the rotation composed of a rotation by angle $\pi / 2$ about the $z$-axis followed by a rotation by angle $\pi / 2$ about the $x$-axis. Determine the rotation axis $\mathbf{n}$ and the magnitude of the angle of rotation $\phi$.

(b) A surface in $\mathbb{R}^{3}$ is given by

$7 x^{2}+4 x y+3 y^{2}+2 x z+3 z^{2}=1 .$

By considering a suitable eigenvalue problem, show that the surface is an ellipsoid, find the lengths of its semi-axes and find the position of the two points on the surface that are closest to the origin.

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