2 and v! = 3 i! + 5 j! are given.

Transcription

1 1. ABCD is a rectangle and O is the midpoint of [AB]. D C 2. The vectors i!, j! are unit vectors along the x-axis and y-axis respectively. The vectors u! = i! + j! 2 and v! = 3 i! + 5 j! are given. (a) Find u! + 2 v! in terms of i! and j!. A O B A vector w! has the same direction as u! + 2 v!, and has a magnitude of 26. (b) Find w! in terms of i! and j!. Express each of the following vectors in terms of OC and OD (a) (b) CD OA (c) AD Answers: (a)... (b)... (Total 4 marks) Answers: (a)... (b)... (c)... (Total 4 marks) 1 2

2 & 6 3. The circle shown has centre O and radius 6. OA is the vector! # & 6, OB is the vector % 0! # and " % 0 " & 5 OC is the vector! #. % 11" y 4. The quadrilateral OABC has vertices with coordinates O(0, 0), A(5, 1), B(10, 5) and C(2, 7). (a) Find the vectors OB and AC. (b) Find the angle between the diagonals of the quadrilateral OABC. C B O A x Answers: (a)... (a) Verify that A, B and C lie on the circle. (b)... (Total 4 marks) (b) Find the vector AC. (c) Using an appropriate scalar product, or otherwise, find the cosine of angle OA ˆ C. (d) Find the area of triangle ABC, giving your answer in the form a 11, where a. (Total 12 marks) 5. The vectors u, v are given by u = 3i + 5j, v = i 2j. Find scalars a, b such that a(u + v) = 8i + (b 2)j.... (Total 4 marks) 3 4

3 6. Find a vector equation of the line passing through ( 1, 4) and (3, 1). Give your answer in the form r = p + td, where t. (c) At this time (06:30) the Chryssault stops and its crew begin their day s work, laying cable in a northerly direction. The Toyundai continues travelling in the same direction at the same speed until it is exactly north of the Chryssault. The Toyundai crew then begin their day s work, laying cable in a southerly direction. At what time does the Toyundai crew begin laying cable? (d) Each crew lays an average of 800 m of cable in an hour. If they work non-stop until their lunch break at 11:30, what is the distance between them at this time?... (Total 4 marks) (e) How long would the Toyundai take to return to base camp from its lunch-time position, assuming it travelled in a straight line and with the same average speed as on the morning journey? (Give your answer to the nearest minute.) (5) (Total 20 marks) & 1 7. In this question, the vector! # & 0 km represents a displacement due east, and the vector % 0! # km a " % 1" displacement due north. Two crews of workers are laying an underground cable in a north south direction across a desert. At 06:00 each crew sets out from their base camp which is situated at the origin (0, 0). One crew is in a Toyundai vehicle and the other in a Chryssault vehicle. & 18 The Toyundai has velocity vector! # km h 1, and the Chryssault has velocity vector % 24" & 36 #! km h 1. % 16" 8. The line L passes through the origin and is parallel to the vector 2i + 3j. Write down a vector equation for L.... (a) Find the speed of each vehicle. (Total 4 marks) (b) (i) Find the position vectors of each vehicle at 06:30. (ii) Hence, or otherwise, find the distance between the vehicles at 06:

4 9. The triangle ABC is defined by the following information & 2 OA =! # & 3, AB = % 3! # & 0, AB BC = 0, AC is parallel to " % 4"! #. % 1 " (b) Write down the vector OC. (a) On the grid below, draw an accurate diagram of triangle ABC. 4 y (b)... (Total 4 marks) 2 1 O x & In this question the vector! # & 0 km represents a displacement due east, and the vector % 0! # km " % 1" represents a displacement due north. The point (0, 0) is the position of Shipple Airport. The position vector r 1 of an aircraft Air One is given by & 16# & 12 r 1 =! #! + t, % 12" % 5" where t is the time in minutes since 12:00. (a) Show that the Air One aircraft (i) is 20 km from Shipple Airport at 12:00; (ii) has a speed of 13 km/min. 7 8

5 (b) Show that a cartesian equation of the path of Air One is: 5x + 12y = 224. The position vector r 2 of an aircraft Air Two is given by & 23 # & 2.5 r 2 =! #! + t, % 5" % 6" & Find the size of the angle between the two vectors! # & 6 and % 2! #. Give your answer to the " % 8 " nearest degree. where t is the time in minutes since 12:00. (c) Find the angle between the paths of the two aircraft.... (d) (i) Find a cartesian equation for the path of Air Two. (Total 4 marks) (ii) Hence find the coordinates of the point where the two paths cross. (5) (e) Given that the two aircraft are flying at the same height, show that they do not collide. (Total 20 marks) & A line passes through the point (4, 1) and its direction is perpendicular to the vector! #. Find % 3" the equation of the line in the form ax + by = p, where a, b and p are integers to be determined.... (Total 4 marks) 9 10

6 & In this question the vector! # & 0 km represents a displacement due east, and the vector % 0! # km " % 1 " represents a displacement due north. The diagram shows the path of the oil-tanker Aristides relative to the port of Orto, which is situated at the point (0, 0) y Path of Aristides 0 1 Not to scale 1 0 & 18 Another ship, the cargo-vessel Boadicea, is stationary, with position vector! # km. % 4" (d) Show that the two ships will collide, and find the time of collision. & 5 To avoid collision, the Boadicea starts to move at 13:00 with velocity vector! # km h 1. % 12 " (e) Show that the position of the Boadicea for t 1 is given by & x # & # & # = 13! + 5! t! % y" % 8" % 12" Orto x (f) Find how far apart the two ships are at 15:00. (Total 20 marks) The position of the Aristides is given by the vector equation & x # & # & # = 0! + 6! t! % y" % 28" % 8" at a time t hours after 12:00. (a) Find the position of the Aristides at 13: Find the angle between the following vectors a and b, giving your answer to the nearest degree. a = 4i 2j b = i 7j (b) Find (i) the velocity vector; (ii) the speed of the Aristides.... (c) Find a cartesian equation for the path of the Aristides in the form (Total 4 marks) ax + by = g. 15. In this question, a unit vector represents a displacement of 1 metre. A miniature car moves in a straight line, starting at the point (2, 0)

7 After t seconds, its position, (x, y), is given by the vector equation & x # & # & # = 2! + 0.7! t! % y" % 0" % 1 " 16. The diagram below shows a line passing through the points (1, 3) and (6, 5). y (a) How far from the point (0, 0) is the car after 2 seconds? (6,5) (b) Find the speed of the car. (1,3) (c) Obtain the equation of the car s path in the form ax + by = c. 0 x Another miniature vehicle, a motorcycle, starts at the point (0, 2), and travels in a straight line with constant speed. The equation of its path is y = 0.6x + 2, x 0. Eventually, the two miniature vehicles collide. (d) Find the coordinates of the collision point. Find a vector equation for the line, giving your answer in the form & x # & # & # = a! + c! t!, where t is any real number. % y" % b " % d " (e) If the motorcycle left point (0, 2) at the same moment the car left point (2, 0), find the speed of the motorcycle. (5) (Total 14 marks)... (Total 4 marks) 13 14

8 17. & 2x The vectors! # & x + 1 and % x 3! # are perpendicular for two values of x. " % 5 " (a) Write down the quadratic equation which the two values of x must satisfy. (b) Find the two values of x. (a) (i) & 0.96# Show that a unit vector in the direction of OA is. 0.28! % " (ii) & v1 # Write down the velocity vector for the airplane in the form!. % v2 " (iii) How long does it take for the airplane to reach A? (5) At A the airplane changes direction so it now flies towards B. The angle between the original direction and the new direction is θ as shown in the following diagram. This diagram also shows the point Y, between A and B, where the airplane comes closest to X. Answers: (a)... (b)... Diagram not to scale X Y B (Total 4 marks) O A 18. The diagram below shows the positions of towns O, A, B and X. Diagram not to scale X B (b) Use the scalar product of two vectors to find the value of θ in degrees. (c) (i) Write down the vector AX. (ii) & 3 Show that the vector n =! # is perpendicular to AB. % 4 " O A (iii) By finding the projection of AX in the direction of n, calculate the distance XY. (6) Town A is 240 km East and 70 km North of O. Town B is 480 km East and 250 km North of O. Town X is 339 km East and 238 km North of O. An airplane flies at a constant speed of 300 km h 1 from O towards A. (d) How far is the airplane from A when it reaches Y? (Total 18 marks) 15 16

9 & x # & 1# & A vector equation of a line is! #! =! + t, t. % y" % 2" % 3 " Find the equation of this line in the form ax + by = c, where a, b, and c. & a The point W has position vector! #, a. % 17" (d) (i) If EW = 2 13, show that one value of a is 3 and find the other possible value of a. (ii) For a = 3, calculate the angle between EW and ET. (10) (Total 19 marks) Calculate the acute angle between the lines with equations & 4 r =! # & 4 + s % 1! # & 2 and r = " % 3! # & 1 + t " % 4! # " % 1" 20. Three of the coordinates of the parallelogram STUV are S( 2, 2), T(7, 7), U(5, 15). (a) Find the vector ST and hence the coordinates of V. (5) (b) Find a vector equation of the line (UV) in the form r = p + λd where λ.... & 1 (c) Show that the point E with position vector! # is on the line (UV), and find the value of % 11" λ for this point

10 22. The following diagram shows the point O with coordinates (0, 0), the point A with position vector a = 12i + 5j, and the point B with position vector b = 6i + 8j. The angle between (OA) and (OB) is θ. y Diagram not to scale C 23. The vector equations of two lines are given below. & 5 r 1 =! # & 3 + λ % 1! # & 2, r 2 = " % 2! # & 4 + t " % 2! # " % 1" The lines intersect at the point P. Find the position vector of P. B A Find (i) a ; O x... (ii) a unit vector in the direction of b; (iii) the exact value of cos θ in the form q p, where, p, q. 24. Consider the vectors c = 3i + 4j and d = 5i 12j. Calculate the scalar product c d.... (Total 2 marks) 19 20

11 & The diagram shows a parallelogram OPQR in which OP =! # & 10#, OQ =. % 3 " 1! % " y P & A vector equation for the line L is r =! # & 3 + t % 4! #. " % 1" Which of the following are also vector equations for the same line L? & 4 A. r =! # & 2 + t % 4! #. " % 1" O R Q x B. & 4 r =! # & 6 + t % 4! #. " % 2" C. & 0 r =! # & 1 + t % 1! #. " % 3" (a) Find the vector OR. 15 (b) Use the scalar product of two vectors to show that cos O Pˆ Q =. 754 & 7 D. r =! # & 3 + t % 5! #. " % 1" (c) (i) Explain why cos P Qˆ R = cos O Pˆ Q. (ii) Hence show that sin P Qˆ R = (iii) Calculate the area of the parallelogram OPQR, giving your answer as an integer. (7) (Total 14 marks)

12 & (a) Find the scalar product of the vectors! # & 30 and % 25! #. " % 40 " (b) Two markers are at the points P (60, 25) and Q ( 30, 40). A surveyor stands at O (0, 0) and looks at marker P. Find the angle she turns through to look at marker Q. Answers: (a)... (b)... (b) & d The position vector of point D is! #. Find d. % 4" (c) Find BD. The line L passes through B and D. (d) (i) Write down a vector equation of L in the form & x # & 1! = % y! # & m# + t. " % 7! " % n " (ii) Find the value of t at point B. (e) Let P be the point (7, 5). By finding the value of t at P, show that P lies on the line L. (1) 28. The diagram shows points A, B and C which are three vertices of a parallelogram ABCD. The & 2# point A has position vector. 2! % " (f) Show that CP is perpendicular to BD. (Total 16 marks) y 10 C B 5 A 5 10 x (a) Write down the position vector of B and of C

13 29. Two lines L 1 and L 2 have these vector equations. L 1 : r = 2i + 3j + t(i 3j) L 2 : r = i + 2j + s(i j) The angle between L 1 and L 2 is θ. Find the cosine of the angle θ... & The points A and B have the position vectors! # & 3 and % 2! # respectively. " % 1" (a) (i) Find the vector AB. (ii) Find AB. & d The point D has position vector! # % 23" (b) Find the vector AD in terms of d. The angle B ÂD is 90. (c) (i) Show that d = 7. (ii) Write down the position vector of the point D. The quadrilateral ABCD is a rectangle. (d) Find the position vector of the point C. (e) Find the area of the rectangle ABCD. (Total 15 marks) 31. Points A, B, and C have position vectors 4i + 2j, i 3j and 5i 5j. Let D be a point on the x- axis such that ABCD forms a parallelogram. (a) (i) Find BC. (ii) Find the position vector of D. (b) Find the angle between BD and AC. (6) 25 26

14 The line L 1 passes through A and is parallel to i + 4j. The line L 2 passes through B and is parallel to 2i + 7j. A vector equation of L 1 is r = (4i + 2j) + s(i + 4j). (c) Write down a vector equation of L 2 in the form r = b + tq. (1) (a) OG ; (b) BD ; (c) EB. (d) The lines L 1 and L 2 intersect at the point P. Find the position vector of P. (Total 15 marks) 32. The diagram shows a cube, OABCDEFG where the length of each edge is 5cm. Express the following vectors in terms of i, j and k. z D E k j F C y G O i B A x 27 28

15 33. A triangle has its vertices at A( 1, 3), B(3, 6) and C( 4, 4). (a) Show that the position vector b of the balloon at time t is given by (a) Show that AB AC = 9 (b) Show that, to three significant figures, cos B ÂC = & x # & 0# & 10.8#!!! b = y! = 0! + t 14.4!.! 5! 0! % z " % " % " At time t = 0, a helicopter goes to deliver a message to the balloon. The position vector h of the helicopter at time t is given by & x # & 49# & 48#!! y! = 32! + t 24 h =!!!!! % z " % 0 " % 6 " (b) (i) Write down the coordinates of the starting position of the helicopter. (ii) Find the speed of the helicopter. (c) The helicopter reaches the balloon at point R. (i) Find the time the helicopter takes to reach the balloon. (6) 34. In this question, distance is in kilometers, time is in hours. A balloon is moving at a constant height with a speed of l8 km h 1, in the & 3#! direction of the vector 4!! % 0" At time t = 0, the balloon is at point B with coordinates (0, 0, 5). (ii) Find the coordinates of R. (5) (Total 15 marks) 29 30

16 & Find the cosine of the angle between the two vectors! # & 2 and % 4! #. " % 1" (a) (i) Find AB. (ii) & 0.8 Show that the vector of length one unit in the direction of AB is! #. % 0.6" 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. & 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 (iii) At what time is the aircraft flying over town B? 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. (c) How far from town C will the aircraft be when the warning light comes on? (6) (7) (Total 17 marks) 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

17 37. 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. (c) The point D is the centre of a circle and E is on the circumference as shown in the following diagram. (c) Write down a vector equation representing the position of B, giving your answer in the form r = a + tb. The point G is also on the circumference. DE is perpendicular to DG. Find the possible coordinates of G. (8) (Total 12 marks) Answers: (a)... (b)... (c) 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 " % 12! #. " % 3" (a) Find the position vector of the car after 2 seconds. 38. Consider the point D with coordinates (4, 5), and the point E, with coordinates (12, 11). (a) Find DE. (b) Find DE. Car 2 moves in a straight line starting at point B (14, 0). Its position q seconds after it starts is & x given by! # & 14 = % y! # & 1 + q " % 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) 33 34

18 40. The points P( 2, 4), Q (3, 1) and R (1, 6) are shown in the diagram below. 41. 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. (a) Find the vector PQ. (b) Find a vector equation for the line through R parallel to the line (PQ). (b) (i) Find the position vector of S. (ii) Write down the equation of L 1. The line L 2 has equation r = (5i +10 j +10k) + s ( 2i + 5 j 3k). (c) 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) 35 36

19 42. The following diagram shows a solid figure ABCDEFGH. Each of the six faces is a parallelogram. (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) (c) Calculate the volume of the solid ABCDEFGH. (d) The coordinates of G are (9, 14, 12). Find the coordinates of H. The coordinates of A and B are A (7, 3, 5), B(17, 2, 5). (e) The lines (AG) and (HB) intersect at the point P. (a) Find (i) AB; (ii) AB. & 2 #! Given that AG = 7!, find the acute angle at P.! % 17" (5) (Total 19 marks) The following information is given. 43. The line L passes through the points A (3, 2, 1) and B (1, 5, 3). & 6# & 2#!! AD = 6!, AD = 9, AE = 4!, AE = 6! % 3! " % 4 " (a) Find the vector AB

20 (b) Write down a vector equation of the line L in the form r = a + tb. 44. 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

21 45. Consider the vectors u = 2i + 3 j k and v = 4i + j pk. (a) Given that u is perpendicular to v find the value of p. (b) Given that q u =14, find the value of q. The point R (2, y 1, z 1 ) also lies on L 1. (b) Find the value of y 1 and of z 1. The line L 2 has equation r = 2i + 9 j +13k + t (i + 2 j + 3k). (c) The lines L 1 and L 2 intersect at a point T. Find the position vector of T. (d) Calculate the angle between the lines L 1 and L In this question, distance is in metres, time is in minutes. Two model airplanes are each flying in a straight line. (7) (7) (Total 22 marks) At 13:00 the first model airplane is at the point (3, 2, 7). Its position vector after t minutes is & x # & 3# & 3 #!!! given by y! = 2! + t 4!.! % z! " % 7! " % 10" (a) Find the speed of the model airplane. 46. Points P and Q have position vectors 5i +11j 8k and 4i + 9 j 5k respectively, and both lie on a line L 1. At 13:00 the second model airplane is at the point ( 5, 10, 23). After two minutes, it is at the point (3, 16, 39). (a) (i) Find PQ. (ii) Hence show that the equation of L 1 can be written as r = ( 5 + s) i + (11 2s) j + ( 8 + 3s) k. (b) & x # & 5# & 4#!!! Show that its position vector after t minutes is given by y! = 10! + t 3!.! % z! " % 23! " % 8" 41 42

22 (c) The airplanes meet at point Q. (i) At what time do the airplanes meet? (ii) Find the position of Q. (6) & 0# & 6#!! (b) An equation for the line (MK) is r = 7! + s 7!.! % 0! " % 10 " (i) Write down an equation for the line (JQ) in the form r = a + tb. (d) Find the angle θ between the paths of the two airplanes. (6) (Total 17 marks) (ii) Find the acute angle between (JQ) and (MK). (c) The lines (JQ) and (MK) intersect at D. Find the position vector of D. (9) (5) (Total 16 marks) 48. The diagram below shows a cuboid (rectangular solid) OJKLMNPQ. The vertex O is (0, 0, 0), J is (6, 0, 0), K is (6, 0, 10), M is (0, 7, 0) and Q is (0, 7, 10). 49. Consider the points A (1, 5, 4), B (3, 1, 2) and D (3, k, 2), with (AD) perpendicular to (AB). (a) Find (i) AB ; (ii) AD, giving your answer in terms of k. (b) Show that k = 7. 1 The point C is such that BC = AD. 2 & 6#! (a) (i) Show that JQ = 7!.! % 10 " (ii) Find MK. (c) Find the position vector of C. (d) Find cos A Bˆ C. (Total 13 marks) 43 44

23 50. Let v = 3i + 4 j + k and w = i + 2 j 3k. The vector v + pw is perpendicular to w. Find the value of p. (Total 7 marks) 51. The point O has coordinates (0, 0, 0), point A has coordinates (1, 2, 3) and point B has coordinates ( 3, 4, 2). & 4#! (a) (i) Show that AB = 6!. 1! % " (ii) Find B ÂO. & x # & 3# & 4#!!! (b) The line L 1 has equation y! = 4! + s 6!.! 2! 1! % z " % " % " Write down the coordinates of two points on L 1. (c) The line L 2 passes through A and is parallel to OB. (i) Find a vector equation for L 2, giving your answer in the form r = a + tb. (ii) Point C (k, k, 5)is on L 2. Find the coordinates of C. & x # & 3 # & 1 #!!! (d) The line L 3 has equation y! = 8! + p 2!, and passes through the point C.! 0! 1! % z " % " % " (8) (6) Find the value of p at C. (Total 18 marks) 45 46

24 & 2# & 1# & 3 # & 1#!!!! 52. The line L 1 is represented by r 1 = 5! + s 2! and the line L 2 by r 2 = 3! + t 3!.!! % 3" % 3 " 8! 4! % " % " The lines L 1 and L 2 intersect at point T. Find the coordinates of T. 47