Problem 3 (Spin tensor) Suppose we have a particle of mass m rotating with angular velocity about some axis. The particle's velocity is given by V = Wxr, where r is the position of the particle with respect to the origin O, on the axis of rotation: (5 points) Without choosing any basis vectors, show that v is linear in r. That is, regarding v as a function of r, show that for any scalars a and b, and any two position vectors r1 and r2, (ar1 + br2) = av(r1) + bv(r2) (10 points) In light of (5), we can immediately deduce that there exists a rank-two tensor (known as the spin tensor) such that V = M.r. Now choose an arbitrary Cartesian basis {i, j, k} in which V = Uxi + Vyj + Wzk. Find the nine components of M in the given basis such that Ux = Mxx, Uy = Mxy, Uz = Mxz, Vx = Myx, Vy = Myy, Vz = Myz, Wx = Mzx, Wy = Mzy, and Wz = Mzz. Express your answers in terms of Wx, Wy, and Wz.