Find the norm of the vector a in the following cases. (a) a = (2, - 1), ft. (-1, 1) (b) a = (-1, 3). b = (0.4) (c) a = (2, -1, 5), b = (-1, 1, 1) (d) a = (- 1, -2, 3) b = (-1, 3, -4) (e) a = (pi, 3, -1), b = (2 pi. -3, 7) (f) a =(15, -2, 4), b = (pi, 3, -1) find the norm of vector b in the above cases. find the projection of a along b in the above cases. find the projection of b along a in the above cases. find the cosine between the following vectors a and b. (a) a = (1, -2) and b = (5, 3) (b) a = (- 3, 4) and b = (2, -1) (c) a =(1, -2, 3) and b = (-3, 1, 5) (d) a =(-2, 1, 4) and b = (-1, -1, 3) (e) a =(-1, 1.0) and b = (2, 1, -1) determine the cosine of the angles of the triangle whose vertices are (a) (2, -1, 1), (1, -3, -5), (3, -4, -4). (c) (3.1, 1), (-1, 2, 1), (2, -2, 5). let a_1, a, be non-zero vectors which are mutually perpendicular, in other words a_i middot a_j = 0 if i notequato j. let c_1, c_r be numbers such that c_1a_1 + middot middot middot c_ra_r = 0. show that all c_i = 0. for any vectors a, b. prove the following relations: (a) || a + b ||^2 + ||a - b||^2 = 1 || a ||^2 + 2 ||b||^2. (d) || a + b ||^2 = ||a||^2 + ||b||^2 + 2a middot b. (c) || a+ b ||^2 - ||a - b||^2 = 4a middot b. interpret (a) as a "parallelogram law".