Problem 1.5: For this problem, we need to figure out the length of the blue segment shown in the figure. This can be solved easily using similar triangles (where the diameter d comes into picture). The second part follows directly from this result, with some algebra.

Problem 2.8: For this problem, we can assume that the x-axes are aligned but not the y-axes. The solution is to figure the linear transform between the two coordinates systems (one with orthogonal axes and one without). In one coordinates, the two basis vectors are (1, 0) and (0, 1). In the other, the basis vectors are (1, 0), (cos θ, sin θ). θ Problem 2.12: Suppose M is the coordinates transform between the two frames, say from A to B. You need to write BΠ in terms of M and AΠ (in fact, in terms of transpose and inverse of M).

Problem 3.1: A generalized eigenvector between two matrices A and B is a vector x such that Ax = λ Bx for some nonzero real number λ. This problem can be done quickly using Lagrange multipliers (in fact, the generalized eigenvalue λ is the Lagrange multiplier). Problem 3.5: This is a simple matrix multiplication once we figure out the matrices for rotations about y and z axes. Problem 3.6: This problem can be solved using geometry. By rotating an angle θ, we mean rotating on a plane perpendicular to the rotation axis U. The rotation axis is of course fixed by the rotation. The projection of the vector X on the U axis is (U \dot X) U and there are two perpendicular vectors on the plane orthogonal to U: X – ( U \dot X) U (=v1) and U x X (v2) If we normalized v1 and v2 to unit vectors, the three unit vectors v1, v2 and U can be used to write down a matrix for the rotation that will give the result immediately. U X U x X