Let P be the plane given by —2x + y + 2z = 5. Let P' be the plane with the same normal vectors as P but through the point Q(2, 0, 4). (a) Find a Cartesian equation for P'. (b) Show that the Cartesian...

Let P be the plane given by
—2x + y + 2z = 5. Let P' be the plane with the same normal vectors as P but through the point Q(2, 0, 4). (a) Find a Cartesian equation for P'. (b) Show that the Cartesian equations for P and P' form an inconsistent system. (c) Let R be the point (2, —1, 2). Calculate the distance from R to P'. (2) (a) Solve the following system of linear equations for x,y,z and w. {x+y+z+w=0 x—y+z—w=1 x+y—z—w=1 x—y—z-Ftv=1 (b) For two square matrices M, N of the same size, we define [M, NI = MN — NM. Let X = [0 ro i 0] i ro l and H = [01 01] . ' Y = — [1 o Oi Calculate constants a, b,c,d,e, f,g,h and i such that the following equations hold: [X,11 = aH + bX + cY, [H, X] = dH + eX + fY, fil Yl= al f 4 hX. 4 iV