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QUESTION 4 Consider points A and B as shown. NP e e e e o o - 10 . X The position vector representing the midpoint of AB is 5 (A) |85 10 5 10 8.5 | : o[ ; (B) 5 10 5 8.5 (D)
QUESTION 8 An equation of the line passing through the points A(2 4 5) and B(3 -2 1) is (A) (B) ©) (D) 2i+4j+5k+t(3i—2j+k) tER —31+2j—k+t(i—6j—4k) t€R x—1 y+6 z+4 2 4 5 -3 y+2 z-1 6 4
QUESTION 12 (8 marks) Consider the vertices A B and C of the rectangular prism as shown. a) State the coordinates of A B and C. [1 mark] b) Determine a unit vector n that is normal to the plane containing A B and C. [3 marks] ¢) Verify that 7 is perpendicular to 4B . [2 marks] d) Determine the Cartesian equation of the plane that contains A B and C. [2 marks]
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 9 Two objects P and Q move in three-dimensional space such that their positions r over time ¢ are described by the following vectors until they collide. re =2 —40)i+ 22—t +3)j— (6 - 5t)k ro = (—t2+20)i + Bt + t)j + t?k The objects will collide at (A) t=0 (B) t=1 C) t=2 (D) t=3
QUESTION 15 (4 marks) The position vectors of points P and Q are 2{ — 3j + k and 2i + 2j — 4k respectively. Let O be the origin. a) Determine the angle POQ. [2 marks] Points O P and Q are joined to form a triangle. b) Determine the area of triangle POQ. [2 marks]
QUESTION 12 (8 marks) Consider the plane x—y—2z=15. a) Determine a vector n that is perpendicular to the plane. [1 mark] b) Determine the vector equation of the line / that is perpendicular to the plane and contains the point A(-2 1 3). [1 mark] c) Use the result from Question 12b) to express the equation of the line / in parametric form. /I mark] The line / and the plane intersect at point S. d) Show that the coordinates of S are (2 -3 - 5). [3 marks] ¢) Determine AS. [1 mark] f) Use a property of parallel vectors to verify that AS and n are parallel. [1 mark]
QUESTION 5 1 1 A vector normal to the plane that contains the vectors | 3 | and | 0 | 1s 0 2 (A) 6i+2j+3k (B) 6i+2j-3k C) 6i-2j+3k (D) 6i-2j-3k
QUESTION 6 The Cartesian equation of a sphere is given by X+ y2 +z2+2x-2 y=1T. The centre and radius of the sphere are (A) (-1 1 0) and 3 respectively. B) (-1 (C) (1 -1 0) and 3 respectively. ( — (D) —1 1 0) and 9 respectively. 1 —1 0) and 9 respectively.
QUESTION 11 (4 marks) OABC is a triangular-based pyramid as shown. A(l 2 5) C(0 5 2) Not to scale Use a vector method to determine the area of the shaded face of the pyramid.
QUESTION 6 The Cartesian equation for a sphere with centre (-2 3 —4) and radius 9 is (A) (x=2) +(y+3) +(z-4) =9
QUESTION 10 A plane is represented by the equation x —2z = 5. A vector normal to this plane is 1 (A) | -2 5 (B) o O = -2 ~ O N 7~ N 7 N -/ O M AN AN _/ -5
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 7 Givena = (3n + 2); + 2} b= (n — 2)} and axb = (1 — 2n)l€ the possible values of n are (A) —-5and 2 3 5 B) —land — (B) 3 (C) land —% (D) 5and —%
QUESTION 15 (5 marks) Consider points A(3 —1 3)and B(1 1 6). a) Determine AB. [1 mark] b) Determine the Cartesian equation of the line that passes through points A and B. [2 marks] Point A lies on the plane ¢ and AB is perpendicular to this plane. c) Determine the Cartesian equation of the plane. [2 marks]
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 2 The line [ 1s described by the vector equation X 0 4 (y)=<7>+k(16) Z 13 —2 Which of the following points lies on line [? A) (0 7 11) (B) (2 15 12) (C) (416 11) (D) (4 23 15)
QUESTION 6 1 The vector (—2) is normal to the plane containing the points (0 0 0) (3 4 5) and (2 k 4). 1 The value of k is (A) —6.5 (B) -3 © 3 (D) 6.5
QUESTION 16 (4 marks) A plane passes through the points A(0 0 1) B(1 2 3) and C(—1 0 —2). a) Determine a vector normal to the plane. [2 marks] b) Use your result from 16a) to determine the Cartesian equation of the plane. [2 marks]
QUESTION 19 (7 marks) Two objects A and B move in three-dimensional space such that their positions over time (t = 0) are described by the following vectors until they collide. r (t) = cos(at) i + 2 sin(at) j — V3 cos(at) k bt V3t rp(t) = i +/3btj +—k a) Given —t < a <0 determine when and where the objects collide. [5 marks] b) Evaluate the reasonableness of your result from 19a). [2 marks]
QUESTION 8 Point A is the centre of a sphere and point B lies on its surface as shown. The equation of the sphere is (A) x7 -2x + y74274+27=23 (B) x7 +2x4+y?4+27-27=23 (C) x? -2x4+y? 4727422 =25 (D) x7 4+2x+y?427-27=25 @ ici 0 -1) @ B(1 -3 -5) Z KRr-c-cccoo oo \
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]
QUESTION 19 (6 marks) Object A is released from the origin with constant velocity v such that its position after t seconds is given by r =2V3tit+3tj+2tk t>0. At a later time object B is released from point P (3V3 6 0) and travels towards point 0(5V3 8 4) with constant velocity vp such that Iv al =/2 lv Al: Given that objects A and B collide determine the time between the release of the two objects. Assume all positions are given in metres and all velocities are given in metres per second.
QUESTION 5 A plane contains the origin and the points (1 2 3) and (3 2 1). A vector normal to the plane is 4 (A) | -8
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