Part B: Angle of rotation Having found the axis of rotation of the satellite, you will now nd its angular velocity from your measure- ments. 1. With the help of a sketch, show geometrically that an anticlockwise rotation by an angle of 6 about the zaxis is represented by the matrix cos(3) sin(9) 0 R2 = sin(9} 003(6) 0 0 D 1 You may use the matrix for a 2-D rotation covered in the lectures. [2 marks] 2. Find all real and complex eigenvalues of R; in terms of 6'. [2 marks] It is a fact that the matrix R for any rotation by an angle of 8 about some axis of rotation can written as R = F'RJ'"l for some invertible matrix P, where P is independent of 6.2 3. Show that if P is an invertible matrix and X is any arbitrary matrix of the same size, then det(X AI) = detI[P)'(P'l Al) You may nd some of the results from the Week 3 applied class problem sheet useful here. [2 marks] 4. Hence explain why R and Rz must have the same eigenvalues, and express the sum of the eigenvalues of R in terms of B. [2 marks] 5. Find the angular velocity of the satellite in radians per second, up to 4 decimal places of accuracy. [2 marks] You notice that there are three pieces of space debris near the satellite. You wonder if they are rotating uniformly around the satellite. You make measurements of the location of each piece of space debris at time D and again at time 1. The locations are related to your student number in a similar fashion to Assignment 1. If your student number is abcdefgh, then at time 0, the rst debris is at position (ed, ef, 9h] + (10, 10, 10), the second is at position (cd,ef,gh)+(5,10,15), and the third at position (ed, ef,gh)+ (20,20, 20). At time 1, the rst debris is at [100,100, 100) (cd,e_f,gh], the second at (105,110, 100] [cd,ef,gh] and the third at (110,105,120) (cd,ef,gh). 6. With the help of Matlab [or otherwise], determine B, the matrix representing the linear transformation taking the positions of each piece of debris at time 0 to their positions at time 1. [2 marks] 7. Does B represent a rotation? Justify your answer. Hint: you may want to look up and use some properties of rotation matrices, perhaps even ones you have just discovered! [2 marks]