2. (a) In computer graphics and robot kinematics the positions of objects are described using homogeneous coordinates rather than Cartesian coordinates. In general, the Cartesian vector r= (y has a corresponding homogeneous vector V=/y -- Z So another row is added to the vector which contains the number 1. In this question we wish to rotate a vector with homogeneous coordinates --- through an angle of radians about an axis parallel to the z axis passing through a point with homogeneous coordinates 0 (0) First the origin of the coordinate system must be moved to the axis of rotation. [1] by translating the vector V, by towards the origin by the vector V This is achieved using the following matrix multiplication 1 0 0 -1 VI 0 1 0 0 0 1 0 0 0 Giving all your working, show that 0 0 0 1 V [3] 0 1 0 0 0 1. (ii) In order to complete the rotation, V, must be multiplied by a product of matrices, which carries out the rotation and translates the origin back to its original position This product is 1001 cos() -sin(0) 0 0 A = 010 o sin() cos(O) 0 0010 0001 Ife = radians, giving all of your working show that 11 -1 0 1 1 1 1 0 0 1 0 o o o 12 (ii) Find V2, the final position of the vector after the rotation where 11 -1 0 1 1 1 0 0 0 VEO 0 1 0 0 0001 A= VIO 0 [ [1] 1 (b) Consider the 2 x 2 rotation matrix which has application to Cartesian coordinate systems: R COS (30) -sin (30) sin (36) Cos (30) In this question you will need to use the trigonometric identity, cos(A) + sin(A) = 1 Together with Euler's relations ela = cos(A) +j sin (A) And e-11 = cos(A) i sin (4) [5] (i) If an eigenvalue of the matrix R is a show that the obeys the characteristic polynomial 12 - 21 cos(38) + 1 = 0 (ii) Show that the eigenvalues of the matrix Rare 12 = 2/38 and 12 = -338 [3] (iii) Show that, if sin (30) # 0, the process of finding an eigenvector corresponding to the eigenvalue. * = e/30 = cos(30) + sin (30) Leads to the following expression (i =})() = (0) (iv) Applying Gaussian elimination find an eigenvector for R [3]