problem 4. [2.Op] Let the vectors p, q, and r are mutually perpendicular. a. Show that Ilp + q + 7112 = 11p112 + 119112 + 115712. b. Further if p, q, and r are all of the same length, show that they will make the same angle with p + q t r.problem 3. 12.5p] Find the vector and parametric equations of the following lines a. The line parallel to 3 - 2 and passing through P(1, -1, 3) b. The line passing through P(3, -1, 4) and Q(1, 0, -2) c. The line passing through P(1, 0, -3) and parallel to the line with parametric equations x = -1 + 3t, y = 2 - t, and z = 3 + 3t. d. Find the point of intersection (if any) of the following pairs of lines. y = - 7 + s - 2 12Problem 2. [2.5p] Suppose that P1 (3, -1, 6) and P2(1, 1, 4) are two points. a. Find the length between Pi and P2. b. Find the vector from P1 to P2. c. Find the length of P1 P2. d. Find the unit vector of P1P2. Justify your answer. e Find a parallel vector to P1 P2. f. Show that |laP1 P2ll = |all|P1 P2ll, when a = 5. 8. If aPIP2 + 0 and a E R, then find that for what values of a the direction of a PIP2 is the same as P1 P2, opposite to P1 P2.problem 1. 13.0p] Let u = AD, v = DC, w = CB and x = BA, as the figure indicates. Assume that ABCD is a parallelogram with AD parallel to BC. a. Explain why AD = BC. b. Explain why x = -v. c. Write AC in terms of u and v. d. Write AC in terms of x and w. e. Find the sums x tu, (x tu) tv, and ((x tu) + v) +w, and express the answers in terms of the vertices of the parallelogram. f. Show geometrically that vector addition is associative; that is, ( utv) + w) = u+ (3+ w) [Hints: Use the triangular rule for addition] Figure 0.1: Parallelogram