MATH 1300 ASSIGNMENT PROBLEMS (UNIT 1) 1. AOB is the diameter of a circle with centre at O and C is any other point on the circle. Denote the vector OA by a and the vector OC by c. C c A [4] (a) Write the vectors AC and CB in terms of the vectors a and c. [6] a O B (b) Use vector methods to show that ACB is a right angle. 2. Let A = (2, 1, 5) and B = (1, 2, 4) be two points in R3. [2] (a) Find the components of the vectors AB and BA . [2] (b) Find the coordinates of the point C if AC CB . [3] (c) The point D = (k, 1, 5) is equidistant from the points A and B. Find the value(s) of k. [3] (d) Find the coordinates of the point X for which AX 2AB . 3. Let u (2,1,3), v (3, 2, 2) and w (5, 1, 4) be three vectors in R3. Find the following. [2] (a) 2u 3v w [2] (b) u v [2] (c) v u [2] (d) projvu [2] (e) sine of the angle between the vectors u and v. 4. Find value(s) for k so that the angle between u (3, k )and v (2, 1) is [3] [3] [4] a) 900 b) 00 c) 450 5. The direction angles of a vector are the three angles , and that the vector makes with the positive X, Y and Z co-ordinates axes. For the vector (2,3,4) find the following: [2] [2] [2] [4] a) cos b) cos c) cos d) cos2 cos2 cos2 Z (2,3,4) Y X Z [10] 6. The four points A(5,0,0), B(0,0,2), C(0,4,0) and D(5,6,3) form a quadrilateral lying on the plane 4 x 5 y 10 z 20 . Find the area of this quadrilateral. B C A X D Y