Ifd@ = [~4,1,1]and b = [5,-3,9], a. determine the angle between a and b. [5 marks] b. proj.a. [2 marks] axb. [3 marks] express either vector in terms of i, j and k. [1 mark] the area of the parallelogram defined by a and b. [2 marks] verify that 8 x b is orthogonal to @ and b. [2 marks] -~ @ oo Find the values of b and if U = 2i + b] +7kand V=ci + 61? +27%K are collinear. [3 marks] If the points A(3, -1, 4), B(3, 1, 3) and C(7, 4, 'I) are the vertices of a triangle, a. calculate the perimeter of AABC. [3 marks] b. Verify that AABC is a right triangle. [2 marks] A force of 200N is applied to wrench in a clockwise direction at 80 degrees to the handle, 10 cm from the center of the bolt. a. Calculate the magnitude of the torque. Include a diagram in your solution. [3 marks] b. In what direction does the torque vector point? [1 mark] i Calculate the volume of the parallelepiped defined by U = [3, -2, 6], V= [5, 0, 1] andw = [7, 6, 3]. [3 marks] If i = (6,4) and = 10 [N28E] a. convert % into geometric form and then determine |i + 7|. [5 marks] b. convert into component form and then determine the true bearing direction of i v. [4 marks] A plane is traveling at 250 km/hr on a heading of N18E into a 35 km/hr wind from N70W. Determine the plane's ground velocity. [5 marks] A line passes through the point (4, 3) with direction vector [1, 5]. a. Determine the parametric equations of the line? [2 marks] b. What point on the line corresponds to the parameter value t = 2? [2 marks] c. Does the line contain the point P (3, 7)? [2 marks] Find a vector equation and the parametric equations of a line through the points A (1,-3,2)and B (9, 2, 0). [6 marks] 10. 11. 12. 13. 14. Write the scalar equation of the line through the point Q (4, 1) with normal [3, 5]. [3 marks] Determine the intersection of the lines below. [4 marks] Line1: x=1+3t,y=5t,z=4t-3 Line 2: [x,y,2z] =[0, -9, -1] + s [-1, 2, -3] Find a vector equation and the parametric equations of the plane that contains the point (3, -5, 1) and is parallel to [x, y, z] = [-5, 2, -5] + t[3, -1, 1] +s[1, 1, 1]. [4 marks] Find a vector and scalar equation of the plane containing the points G (4, 1, -1), H (0,1,2),and | (1, 1, 1). [5 marks] Determine if the line and the plane intersect. If so, determine the point(s) of intersection. [3 marks] Plane 1: [x, , Z] = [4, 6, 0] + t [-1, 2, 1] Plane 2: 2x -y +4z+10=0