Find a vector equation for the line through R parallel to the line (PQ) (Total 6 marks)
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1 1. The points P( 2, 4), Q (3, 1) and R (1, 6) are shown in the diagram below. (a) Find the vector PQ. (b) Find a vector equation for the line through R parallel to the line (PQ). 2. The position vector of point A is 2i + 3 j + k and the position vector of point B is 4i 5 j + 21k. (a) (i) Show that AB = 2i 8 j + 20k. (ii) Find the unit vector u in the direction of AB. (iii) Show that u is perpendicular to OA. (6) Let S be the midpoint of [AB]. The line L 1 passes through S and is parallel to OA. 1
2 (b) (i) Find the position vector of S. (ii) Write down the equation of L 1. (4) The line L 2 has equation r = (5i +10 j +10k) + s ( 2i + 5 j 3k). Explain why L 1 and L 2 are not parallel. (d) The lines L 1 and L 2 intersect at the point P. Find the position vector of P. (7) (Total 19 marks) 3. The following diagram shows a solid figure ABCDEFGH. Each of the six faces is a parallelogram. The coordinates of A and B are A (7, 3, 5), B(17, 2, 5). (a) Find (i) AB; (ii) AB. (4) The following information is given. 6 AD = 6, AD = 9, AE = 3 2 4, AE = 6 4 2
3 (b) (i) Calculate AD AE. (ii) Calculate AB AD. (iii) Calculate AB AE. (iv) Hence, write down the size of the angle between any two intersecting edges. (5) Calculate the volume of the solid ABCDEFGH. (d) The coordinates of G are (9, 14, 12). Find the coordinates of H. (3) (e) The lines (AG) and (HB) intersect at the point P. 2 Given that AG = 7, find the acute angle at P. 17 (5) (Total 19 marks) 4. The line L passes through the points A (3, 2, 1) and B (1, 5, 3). (a) Find the vector AB. (b) Write down a vector equation of the line L in the form r = a + tb. 3
4 5. The line L passes through A (0, 3) and B (1, 0). The origin is at O. The point R (x, 3 3x) is on L, and (OR) is perpendicular to L. (a) Write down the vectors AB and OR. (b) Use the scalar product to find the coordinates of R Find the cosine of the angle between the two vectors and. 4 1 Working: Answer:
5 1 7. In this question the vector represents a displacement of 1 km east, 0 0 and the vector represents a displacement of 1 km north. 1 The diagram below shows the positions of towns A, B and C in relation to an airport O, which is at the point (0, 0). An aircraft flies over the three towns at a constant speed of 250 km h 1. y B O x A C Town A is 600 km west and 200 km south of the airport. Town B is 200 km east and 400 km north of the airport. Town C is 1200 km east and 350 km south of the airport. (a) (i) Find AB. 0.8 (ii) Show that the vector of length one unit in the direction of AB is. 0.6 (4) An aircraft flies over town A at 12:00, heading towards town B at 250 km h 1. p Let be the velocity vector of the aircraft. Let t be the number of hours in flight after 12:00. q The position of the aircraft can be given by the vector equation x 600 p = + t. y 200 q (b) (i) 200 Show that the velocity vector is. 150 (ii) Find the position of the aircraft at 13:00. (iii) At what time is the aircraft flying over town B? (6) 5
6 Over town B the aircraft changes direction so it now flies towards town C. It takes five hours to travel the 1250 km between B and C. Over town A the pilot noted that she had litres of fuel left. The aircraft uses 1800 litres of fuel per hour when travelling at 250 km h 1. When the fuel gets below 1000 litres a warning light comes on. How far from town C will the aircraft be when the warning light comes on? (7) (Total 17 marks) 8. A boat B moves with constant velocity along a straight line. Its velocity vector is given by 4 v =. 3 At time t = 0 it is at the point ( 2, 1). (a) Find the magnitude of v. (b) Find the coordinates of B when t = 2. Write down a vector equation representing the position of B, giving your answer in the form r = a + tb. Working: Answers: (a)... (b) Consider the point D with coordinates (4, 5), and the point E, with coordinates (12, 11). (a) Find DE. (b) Find DE. 6
7 The point D is the centre of a circle and E is on the circumference as shown in the following diagram. The point G is also on the circumference. DE is perpendicular to DG. Find the possible coordinates of G. (8) (Total 12 marks) 10. Car 1 moves in a straight line, starting at point A (0, 12). Its position p seconds after it starts is x given by 0 = y 5 + p (a) Find the position vector of the car after 2 seconds. Car 2 moves in a straight line starting at point B (14, 0). Its position q seconds after it starts is x 14 1 given by = + q. y 0 3 Cars 1 and 2 collide at point P. (b) (i) Find the value of p and the value of q when the collision occurs. (ii) Find the coordinates of P. (6) (Total 8 marks) 7
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