Vectors: Fundamentals and Problems

CPP FIITJEE ALLEN Vectors PROBLEMS BASED OF FUNDAMENTALS OF VECTOR 1. How many minimum number of coplanar vectors having different m...

CPP FIITJEE ALLEN Vectors PROBLEMS BASED OF FUNDAMENTALS OF VECTOR 1. How many minimum number of coplanar vectors having different magnitudes can be added to give zero resultant? (A) 2 (B) 3 (C) 4 (D) 5 2. A hall has t e dimensions 10m x 12 m 14m. A fly starting at one corner ends up at a diametrically opposite corner. What is the magnitude of its displacement? (A) 17m (B) 26m (C) 36m (D) 21m 3. 0.4iˆ  0.8 ˆj  ckˆ represents a unit vector when c is (A) -0.2 (B) 0.2 (C) 0.8 (D) 0 4. 100 coplanar forces each equal to 10N act on a body. Each force makes angle  / 50 with the preceding force. What is the resultant of the forces? (A) 1000 N (B) 500 N (C) 250 N (D) 0 5. The magnitude of a given vector with ends point (4,-4,0) and (-2,-2,0) must be (A) 6 (B) 5 2 (C) 4 (D) 2 10 6. The angles which a vector ˆi  ˆj  2kˆ makes with X, Y and Z axes respectively are (A) 600,600, 600 (B) 450,450, 450 0 0 0 (C) 60 ,60 , 45 (D) 450,450, 600  1 ˆ 1 ˆ 7. The expression  i j  is a  2 2  (A) Unit Vector (B) Null Vector (C) Vector of magnitude 2 (D) Scalar  8. Given vector A  2iˆ  3ˆj , the angle between A and y- axis is (A) tan-1 3/2 (B) tan-1 2/3 -1 (C) sin 2/3 (D) cos-1 2/3 9. The unit vector along ˆi  ˆj is (A) k̂ (B) ˆi  ˆj i j ˆi  ˆj (C) (D) 2 2 10. A vector is represented by 3iˆ  ˆj  2kˆ. It length in XY plane is (A) 2 (B) 14 (C) 10 (D) 5 FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000 11. Five equal forces of 10 N each are applied at one point and all are lying in one plane. If the angles between then are equal, the resultant force will be (A) Zero (B) 10 N (C) 20 N (D) 10 2N  12. The angle made by the vector A  ˆi  ˆj with x – axis is (A) 900 (B) 450 0 (C) 22.5 (D) 300 13. The value of a unit vector in the direction of vector A  5iˆ  12ˆj , is (A) î (B) ĵ   (C) ˆi  ˆj /13   (D) 5iˆ  12ˆj /13 14. (Multi correct question) Any vector in an arbitrary direction can always be replaced by two(or three) (A) Parallel vectors which have the original vector as their resultant (B) Mutually perpendicular vectors which have the original vector as their resultant (C) Arbitrary vectors which have the original vector as their resultant (D) It is not possible to resolve a vector 15. Angular momentum is (A) A scalar (B) A polar vector (C) an axial vector (D) None of these  16. If a vector P making angles , and  respectively with the X,Y and Z axes respectively. Then sin 2 + sin2 + sin2 = (A) 0 (B) 1 (C) 2 (D) 3 Problem based on addition of vector 17. Two forces, each of magnitude. F have a resultant of the same magnitude F. The angle between the two forces is (A) 450 (B) 1200 (C) 1500 (D) 600 18. For the resultant of the two vectors to be maximum, what must be the angle between them (A) 00 (B) 600 (C) 900 (D) 1800 19. A particle is simultaneously acted by two forces equal to 4N and 3N. The net force on the particle is (A) 7N (B) 5N (C) 1N (D) Between 1N and 7N  20. Two vector A and B lie in a plane, another vector C lies outside this plane, then the    resultant of these three vectors i.e. A  B  C (A) Can be Zero (B) Cannot be Zero     (C) Lies in the plane containing A  B (D) Lies in the plane containing A  B 21. If the resultant of the two forces has a magnitude smaller than the magnitude of larger force, the two forces must be (A)Different in magnitude and have obtuse angle between them (B) Mutually perpendicular to one another (C)Possess extremely small magnitude (D) Having acute angle between them 22. Forces F1 and F2 act on a point mass in two mutually perpendicular directions.The resultant force on the point mass will be (A) F1 + F2 (B) F1  F2 (C) F12  F22 (D) F12  F22 C    B 23. Find the resultant of three vectors OA,OB and OC shown in the following figure. Radius of the circle is R. 0   45 (B) R 1  2 0 (A) 2 R 45 A O (C) R 2 (D) R  2  1       24. If | A  B || A || B | , the angle between A and B is (A) 600 (B) 00 0 (C) 120 (D) 900 25. At what angle must the two forces (x+y) and (x-y) act so that the resultant may be x 2  y2   x2  y2   2(x 2  y 2 )  (A) cos1   2  (B) cos1     2(x  y )  x2  y 2  2   x2  y 2   x2  y2  (C) cos1   2 2  (D) cos1   2   x y   x y  2   26. Let the angle between two nonzero vectors A and B be 1200 and resultant be C     (A) C must be equal to | A  B | (B) C must be less than | A  B |     (C) C must be greater than | A  B | (D) C may be equal to | A  B | E D 27. Fig. shows ABCDEF as a regular hexagon. What is the value      AB  AC  AD  AE  AF   (A) AO (B) 2 AO F C   O (C) 4 AO (D) 6 AO A B       28. The magnitude of vector A,B and C are respectively 12,5 and 13 units and A  B  C then   the angle between A and B is (A) 0 (B)  (C)  / 2 (D)  / 4 29. Magnitude of vector which comes on addition of two vectors, 6iˆ  7ˆj and 3iˆ  4ˆj is (A) 136 (B) 13.2 (C) 202 (D) 160 30. A particle has displacement of 12m towards east and 5m towards north then 6m vertically upward. The sum of these displacements is (A) 12 (B) 10.04 m (C) 14.31 m (D) None of these FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000     31. The Three vectors A  3iˆ  2ˆj  k, ˆ B  ˆi  3ˆj  5kˆ and C  2iˆ  ˆj  4kˆ form (A) An equilateral triangle (B) Isosceles triangle (C) A right angled triangle (D) No triangle 32. For the fig.        C (A) A  B  C (B) B  C  A        B (C) C  A  B (D) A  B  C  V  A    33. Let C  A  B then       (A) | C | is always greater then | A | (B) It is possible to have | C || A | and | C || B | (C) C is always equal to A + B (D) C is never equal to A + B   34. The value of the sum of two vectors A and B with  as the angle between them is (A) A 2  B2  2ABcos  (B) A 2  B2  2ABcos  (C) A 2  B 2  2AB sin  (D) A 2  B 2  2AB sin  35. Following forces start acting on particle at rest at the origin of the co-ordinate system simultaneously F1  4iˆ  5ˆj  5k, ˆ F  5iˆ  8 ˆj  6k, 2 ˆ F  3iˆ  4ˆj  7k, 3 ˆ and F  2iˆ  3 ˆj  2k, 4 ˆ then the particle will move (A) In x – y plane (B) In y – z plane (C) in x - z plane (D) Along x – axis 36. Following sets of three forces act on a body. Whose resultant cannot be zero (A) 10, 10, 10 (B) 10, 10, 20 (C) 10, 20, 20 (D) 10, 20, 40 37. When three forces of 50N, 30N and 15N act on a body, then the body is (A) at rest (B) Moving with a uniform velocity (C) in equilibrium (D) Moving with an acceleration 38. The sum of two forces acting at a point is 16N. If the resultant force is 8N and Its direction is perpendicular to minimum force then the force are (A) 6N and 10N (B) 8N and 8N (C) 4N and 12N (D) 2 N and 14N    39. If vectors P,Q and R have magnitude 5, 12 and 13 units and P  Q  R , the angle between Q and R is 5 5 (A) cos1 (B) cos 1 12 13 1 12 1 7 (C) cos (D) cos 13 13 40. The resultant of two vectors A and B is perpendicular to the vector A and its magnitude is equal to half the magnitude of vector B. The angle between A and B is (A) 1200 (B) 1500 0 (C)135 (D) None FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000 41. What vector must be added to the two vectors ˆi  2ˆj  2kˆ and 2iˆ  ˆj  kˆ , so that the resultant may be a unit vector along x - axis (A) 2iˆ  ˆj  kˆ (B) 2iˆ  ˆj  kˆ (C) 2iˆ  ˆj  kˆ (D) 2iˆ  ˆj  kˆ 42. What is the angle between P and the resultant of P  Q and P  Q     (A) Zero (B) tan-1 P/Q (C) tan-1 Q/P (D) tan-1 (P-Q)/ (P+Q) 43. The resultant of P and Q is perpendicular to P. What is the angle between P and Q (A) cos1(P / Q) (B) cos1(P / Q) (C) sin1 P / Q  (D) sin1  P / Q  44. Maximum and minimum magnitudes of the resultant of two vectors of magnitudes P and Q are in the ratio 3 : 1. Which of the following relations is true? (A) P = 2Q (B) P = Q (C) PQ = 1 (D) None of these 45. The resultant of A  B is R1. On reversing the vector B , the resultant becomes R 2 , what is the value of R12  R22 (A) A2 + B2 (B) A2 - B2 (C) 2(A2 + B2) (D) 2(A2 - B2) 46. The resultant of two vectors P and Q is R. if Q is doubled, the new resultant is perpendicular to P. Then R equals (A) P (B) (P+Q) (C) Q (D) (P-Q) 47. Two forces F1 and F2 are acting on a body. One force is double that of the other force and the resultant is equal to the greater force. Then the angle between the two forces is (A) cos-1(1/2) (B) cos-1(-1/2) -1 (C) cos (-1/4) (D) cos-1(1/4) 48. Given the A  B  C and that C is  to A. Further if | A || C | , then what is the angle between A and B   (A) radian (B) radian 4 2 3 (C) radian (D)  radian 4 FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000  Problems based on subtraction of vectors 49. Figure below shows a body of mass M moving with the uniform P2 speed on a circular path of radius R. What is the change in v acceleration in going from P1 to P2 2 (A) Zero (B) v / 2R v2 P1 (C) 2v 2 / R (D) x 2 R R 50. ˆ F  6ˆj , the third A body is at rest under the action of three forces, two of which are F1  4i, 2 force is (A) 4iˆ  6ˆj (B) 4iˆ  6ˆj (C) 4iˆ  6ˆj (D) 4iˆ  6ˆj 51. A plane is revolving around the earth with a speed of 100 km/hr at a constant height, from the surface of earth. The change in velocity as it travels half circle is (A) 200 km/hr (B) 150 km/hr (C) 100 2 km / hr (D) 0 52. What displacement must be added to the displacement 25iˆ  6ˆj m to give a displacement of 7.0 m pointing in the x – direction? (A) 18iˆ  6ˆj (B) 32iˆ  13ˆj (C) 18iˆ  6ˆj (D) 25iˆ  13ˆj 53. A body moves due East with velocity 20 km/ hr and then due North with velocity 15 km/ hour. The resultant velocity (A) 5 km/ hr (B) 15 km/ hr (C) 20 km/ hr (D) 25 km/ hr P 54. A particle is moving on a circular path of radius r with uniform v velocity v. The change in velocity when the particle moves from P to Q is ( POQ  400 ) O 400 (A) 2v cos 400 (B) 2v sin 200 0 (C) 2v sin 20 (D) 2v cos200 Q v 55. The length of seconds hand in watch is 1 cm. The change in velocity of its tip in 15 seconds is  (A) Zero (B) cm / sec 30 2   2 (C) cm / sec (D) cm / sec 30 30 FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000 56. A particle moves towards east with velocity 5 m/s. After 10 seconds its direction changes towards north with same velocity. The average acceleration of the particle is 1 (A) Zero (B) m / s2 N  W 2 1 1 (C) m / s2 N  E (D) m / s2 S  W 2 2 Problems based on scalar product of vectors 57 Consider two vectors F1  2 i  5k and F2  3 j  4k. The scalar product of these vectors is (A) 20 (B) 23 (C) 5 33 (D) 26 58. Consider a vector F  4iˆ  3ˆj. Another vector that is perpendicular to F is (A) 4iˆ  3ˆj (B) 6iˆ (C) 7kˆ (D) 3iˆ  4ˆj   59. Two vectors A and B are at right angles to each other, when     (A) A  B  0 (B) A  B  0     (C) A  B  0 (D) A.B  0      60. If | V1  V2 |  | V1  V2 | and V2 is finite, then     (A) V1 is parallel to V2 (B) V1  V2     (C) V1 and V2 are mutually perpendicular (D) | V1 |  | V2 |  61.   A force F  5iˆ  3ˆj Newton is applied over a particle which displaces it from its origin to the   point r̂  2iˆ  1jˆ metres. The work done on the particle is (A) -7 joules (B) +13 joules (C) +7 joules (D) +11 joules 62. The angle between two vectors 2iˆ  3ˆj  kˆ and ˆi  2jˆ  4kˆ is (A) 00 (B) 900 0 (C) 180 (D) None of the above 63.   The angle between the vector ˆi  ˆj and ˆj  kˆ is   0 (A) 30 (B) 450 (C) 600 (D) 900 64. A particle moves with a velocity 6iˆ  4ˆj  3kˆ m / s under the influence of a constant force  F  20iˆ  15 ˆj  5kN ˆ. The instantaneous power applied to the particle is (A) 35 J/s (B) 45 J/s (C) 25 J/s (D) 195 J/s     65. If P.Q  PQ, then angle between P and Q is (A) 00 (B) 300 0 (C) 45 (D) 600 FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000    66. Two constant forces F1  2iˆ  3ˆj  3kˆ N and F2  ˆi  ˆj  2kˆ N act on a body and displace it   from the position r1  ˆi  2ˆj  2kˆ  m  to the position r2  7iˆ  10 ˆj  5kˆ  m . What is the work done (A) 9J (B) 41J (C) -3J (D) None of these      67. A force F  5 i  6 j  4k acting on a body, produces a displacement s  6iˆ  5kˆ. Work done by the force is (A) 10 units (B) 18 units (C) 11 units (D) 5 units   68. The angle between the two vector A  5iˆ  5ˆj and B  5iˆ  5ˆj will be (A) 0 (B) 450 0 (C) 90 (D) 1800   69. The vector P  aiˆ  ajˆ  3kˆ and Q  aiˆ  2ˆj  kˆ are perpendicular to each other. The positive value of a is (A) 3 (B) 4 (C) 9 (D) 13 70. A body, constrained to move in the Y direction is subjected to a force given by    F  2iˆ  15ˆj  6kˆ N. What is the work done by this force in moving the body a distance 10m along the Y – axis (A) 20J (B) 150 J (C) 160 J (D) 190 J  71. A particle moves in the x-y plane under the action of a force F such that the value of its      linear momentum P at anytime t is Px = 2 cost, py = 2 sint. The angle  between F and P at a given time t, will be (A)  = 00 (B)  = 300 (C)  = 900 (D)  = 1800 Problem based on cross product of vectors    72 The area of the parallelogram represented by the vectors A  2 i  3ˆj and B  ˆi  4ˆj is (A) 14 units (B) 7.5 units (C) 10 units (D) 5 units         73. For any two vectors A and B if A.B | A  B | , the magnitude of (A  B) is equal to (A) A 2  B2 (B) A+B AB (C) A 2  B2  (D) A 2  B2  2  AB 2   74. A vector F1 is along the positive X-axis. If its vector product with another vector F2 is zero  then F2 could be (A) 4ˆj (B)  ˆi  ˆj    (C) ˆj  kˆ  (D)  4iˆ  FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000      75. If for two vector A and B, A  B  0, the vectors (A) Are perpendicular to each other (B) Are parallel to each other (C) Act at an angle of 600 (D) Act at an angle of 300     76.  The angle between vector A  B and B  A is   (A) Zero (B)  (C)  / 4 (D)  / 2     77.  What is the angle between P  Q and P  Q     (A) 0 (B) 2  (C) (D)  4 78. The resultant of the two vectors having magnitude 2 and 3 is 1. What is the magnitude of their cross product (A) 6 (B) 3 (C) 1 (D) 0   79. Which of the following is the unit vector perpendicular to A and B ˆ  Bˆ A Aˆ  Bˆ (A) (B)  sin  AB AB cos  A B A B (C) (D) AB sin  AB cos    80. Let A  ˆiA cos   ˆjA sin  be any vector. Another vector B which is normal to A is (A) ˆiB cos   ˆjB sin  (B) ˆiB sin   ˆjB cos  (C) ˆiB sin   ˆjB cos  (D) ˆiB cos   ˆjB sin  81. The angle between two vectors given by 6iˆ  6ˆj  3kˆ and 7iˆ  4ˆj  4kˆ is  1   5  (A) cos1   (B) cos1    3  3  2   5 (C) sin1   (D) sin1    3  3       82. A vector A points vertically upward and B points towards north. The vector product A  B is (A) Zero (B) Along west (C) Along east (D) Vertically downward 83.   Angle between the vectors ˆi  ˆj and ˆj  kˆ is   0 (A) 90 (B) 00 (C) 1800 (D) 600   84 Two vector P  2iˆ  bjˆ  2kˆ and Q  ˆi  ˆj  kˆ will be parallel if (A) b = 0 (B) b = 1 (C) b = 2 (D) b = -4 FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000 85 The position vectors of points A,B,C and D are     A  3iˆ  4ˆj  5k, ˆ B  4iˆ  5ˆj  6k, ˆ C  7iˆ  9ˆj  3kˆ and D  4iˆ  6ˆj then the displacement vectors AB and CD are (A) Perpendicular (B) Parallel (C) Antiparallel (D) Inclined at an angle of 600   86. Which of the following is not true? If A  3iˆ  4ˆj and B  6iˆ  8ˆj where A and B are the   magnitudes of A and B respectively A 1 (A) A  B  0 (B)    B 2 (C) A.B  48 (D) A = 5   87.  ˆ ˆ If force F  4i  5 j and displacement (S)  3iˆ  6kˆ then the work done is (A) 4 x 3 (B) 5 x 6 (C) 6 x 3 (D) 4 x 6       88. If | A  B || A.B | , then angle between A andB will be (A) 300 (B) 450 0 (C) 60 (D) 900 89. Choose the correct option. (A) ˆj  kˆ  ˆi (B) ˆi.iˆ  0 (C) ˆj  ˆj  1 ˆ ˆ 1 (D) k.j     90. The linear velocity of a rotating body is given by v    r, where  is the angular velocity and   r is the radius vector. The angular velocity of a body is   ˆi  2ˆj  2kˆ and the radius vector  ˆ then | v | is r  4ˆj  3k, (A) 29 units (B) 31 units (C) 37 units (D) 41 units       91. Three vectors a,b and c satisfy the relation a.b  0 and a.c  0. The vector a is parallel to   (A) b (B) c    (C) b.c (D) b  c 92. The diagonals of a parallelogram are 2iˆ and 2ˆj. What is the area of the parallelogram (A) 0.5 units (B) 1 units (C) 2 units (D) 4 units 93. What is the unit vector perpendicular to the following vectors 2iˆ  2ˆj  kˆ and 6iˆ  3ˆj  2kˆ ˆi  10ˆj  18kˆ ˆi  10ˆj  18kˆ (A) (B) 5 17 5 17 ˆi  10ˆj  18kˆ ˆi  10ˆj  18kˆ (C) (D) 5 17 5 17 94. The area of the parallelogram whose side are represented by the vectors ˆj  3kˆ and ˆi  2ˆj  kˆ (A) 61 sq. units (B) 59 units (C) 49 units (D) 52 units FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000   95.    The position of a particle is given by r  ˆi  2jˆ  kˆ momentum P  3iˆ  4ˆj  2kˆ. The   angular momentum is perpendicular to (A) x – axis (B) y – axis (C) z – axis (D) Line at equal angles to all the three axes     96.      Two vector A and B have equal magnitudes. Then the vector A  B is perpendicular to  (A) A  B (B) A  B   (C) 3A  3B (D) All the these         97. Find the torque of a force F  3 i  j  5k acting at the point r  7 i  3 j  k (A) 14iˆ  38ˆj  16kˆ (B) 4iˆ  4 ˆj  6kˆ (C) 21iˆ  4ˆj  4kˆ (D) 14iˆ  34j  16kˆ     98.    The value of A  B  A  B is  (A) 0 (B) A2 - B2     (C) B  A  (D) 2 B  A  99. A particle of mass m= 5 is moving with a uniform sped v  3 2 in the XOY plane along the line Y = X + 4. The magnitude of the angular momentum of the particle about the origin is (A) 60 units (B) 40 2 units (C) Zero (D) 7.5 units FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000  ANSWERS Vectors 1. B 2. D 3. B 4. D 5. D 6. C 7. A 8. B 9. C 10. C 11. A 12. B 13. D 14. ABC 15. C 16. C 17. B 18. A 19. D 20. B 21 A 22. C 23 B 24. A 25. A 26. B 27. D 28. C 29. C 30. C 31. C 32. C 33. B 34. A 35. B 36. D 37. D 38. A 39. C 40. B 41. B 42. A 43. B 44. A 45. C 46. C 47. C 48. C 49. D 50. D 51. A 52. C 53. D 54. C 55. D 56. B 57. A 58. C 59. D 60. C 61. C 62. B 63. C 64. B 65. A 66. A 67. A 68. C 69. A 70. B 71. C 72. D 73. D 74. D 75. B 76. B 77. B 78. D 79. C 80. C 81 D 82. B 83 D 84. C 85. C 86. C 87. A 88. B 89. A 90. A 91. D 92. C 93. C 94. B 95. A 96. A 97. A 98. D 99. A FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000