Paper 1, Section II, B

Vectors and Matrices | Part IA, 2018

(a) Consider the matrix

R=(cosθsinθ0sinθcosθ0001)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 zz-axis through an angle θ\theta.

Show that RR has three eigenvalues in C\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 π/2\pi / 2 about the zz-axis followed by a rotation by angle π/2\pi / 2 about the xx-axis. Determine the rotation axis n\mathbf{n} and the magnitude of the angle of rotation ϕ\phi.

(b) A surface in R3\mathbb{R}^{3} is given by

7x2+4xy+3y2+2xz+3z2=1.7 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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