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QUESTION 15 (6 marks) The points O(0 0 0) A(-6 2 -2) and C(3 1 2) are represented in three-dimensional space in the diagram. Not drawn to scale z OABC forms a parallelogram in three-dimensional space. a) Determine the coordinates of B. [1 mark] M is the midpoint of BC. b) Determine the vector that represents OM. [1 mark] N divides AM in the ratio 2:1. c) Determine the vector that represents ON. [2 marks] d) Use a vector method to show that O B and N lie on a straight line. [2 marks]
QUESTION 7 The altitude angle of OA is represented as . z A Not to scale X Given the coordinates of A are (3 4 6) the altitude angle of OA in radians is A) 093 (B) 0.88 ) 0.69 D) 0.66
QUESTION 7 Two forces act concurrently on a 2 kg object placed at the origin. y Not to scale The magnitude of the acceleration of the object is (A) 2m 572 (B) 23 ms™ (C) 4ms™> (D) 43 ms™
QUESTION 11 (6 marks) The position vector of a particle r|(cm) over time ¢(s) is given by r ()= (2t +1)i +(¢+3) j— (2t -3)k a) Determine the velocity vector of the particle. [1 mark] b) Determine the time when the position vector of the particle is perpendicular to its velocity vector. [2 marks] The position vector of a second particle rz(cm) over time t(s) is given by ry(t)=(16—4t)i —(3t—13)j+ 2k c) Determine whether the two particles collide. [3 marks]
QUESTION 4 The following diagram shows a vector 04 in three dimensions and its projection OF in the xy plane. Point C lies on the z-axis. 3 X The angle that represents the altitude of the vector 04 is (A) <«AOB (B) <«AOC (C) «£BAO (D) «BOC
QUESTION 11 (9 marks) A parallelepiped is a three-dimensional figure defined by three vectors that form six faces that are parallelograms. F A D Letfizp m=q fi=r. a) Express the area of parallelogram ABGH in terms of p q and/or 7. [1 mark] b) Assuming A is the origin express the vector equation of the line passing through points B and D in terms of p q and/or 7. [2 marks] Two internal diagonals of the parallelepiped AF and BE are contained by the parallelogram ABFE. c) Express both AF and BE in terms of p qandr. [2 marks] d) Use the results from 11c¢) to prove that internal diagonals AF and BE bisect each other. [4 marks]
QUESTION 14 (6 marks) Consider a cube with three edges positioned along the x- y- and z-axes on the Cartesian plane as shown. Points O A and B are vertices of the cube. Not to scale 0% A a) Given OA =2i determine OB. Express your answer in terms of j and k [1 mark] b) Calculate OA x OB. [1 mark] Consider the triangle formed by joining points O A and B. c) Use the result from Question 14b) to determine the area of the triangle. [2 marks] Let points M and N be the midpoints of the triangle’s sides OA and OB respectively. d) Determine MN. [1 mark] e) Use the result from Question 14d) to show that the length of AB is twice the length of MN. [7 mark]
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