Title 01. Vectors Easy.pdf ✅

1. The vector projection of a vector i k ˆ 4 ˆ 3 + on y-axis is (a) 5 (b) 4 (c)3 (d) Zero 2. Position of a particle in a rectangular-co-ordinate system is (3, 2, 5). Then its position vector will be (a) i j k ˆ 2 ˆ 5 ˆ 3 + + (b) i j k ˆ 5 ˆ 2 ˆ 3 + + (c) i j k ˆ 2 ˆ 3 ˆ 5 + + (d) None of these 3. If a particle moves from point P (2,3,5) to point Q (3,4,5). Its displacement vector be (a) i j k ˆ 10 ˆ ˆ + + (b) i j k ˆ 5 ˆ + ˆ + (c) i j ˆ + ˆ (d) i j k ˆ 6 ˆ 4 ˆ 2 + + 4. A force of 5 N acts on a particle along a direction making an angle of 60° with vertical. Its vertical component be (a) 10 N (b) 3 N (c) 4 N (d) 2.5 N 5. If A i j ˆ 4 ˆ = 3 + and , ˆ 24 ˆ B = 7i + j the vector having the same magnitude as B and parallel to A is (a) i j ˆ 20 ˆ 5 + (b) i j ˆ 10 ˆ 15 + (c) i j ˆ 15 ˆ 20 + (d) i j ˆ 20 ˆ 15 + 6. Vector A makes equal angles with x, y and z axis. Value of its components (in terms of magnitude of A ) will be (a) 3 A (b) 2 A (c) 3 A (d) A 3 7. If A i j k ˆ 5 ˆ 4 ˆ = 2 + − the direction of cosines of the vector A are (a) 45 5 and 45 4 , 45 2 − (b) 45 3 and 45 2 , 45 1 (c) 45 4 ,0 and 45 4 (d) 45 5 and 45 2 , 45 3 8. The vector that must be added to the vector i j k ˆ 2 ˆ 3 ˆ − + and i j k ˆ 7 ˆ 6 ˆ 3 + − so that the resultant vector is a unit vector along the y-axis is (a) i j k ˆ 5 ˆ 2 ˆ 4 + + (b) i j k ˆ 5 ˆ 2 ˆ − 4 − + (c) i j k ˆ 5 ˆ 4 ˆ 3 + + (d) Null vector 9. 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 10. A hall has the dimensions 10 m 12 m 14 m. A fly starting at one corner ends up at a diametrically opposite corner. What is the magnitude of its displacement (a) 17 m (b) 26 m (c) 36 m (d) 20 m 11. 100 coplanar forces each equal to 10 N 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) Zero 12. The magnitude of a given vector with end points (4, – 4, 0) and (– 2, – 2, 0) must be (a) 6 (b) 5 2 (c) 4 (d) 2 10 13. The expression         i + j ˆ 2 1 ˆ 2 1 is a (a) Unit vector (b)Null vector (c)Vector of magnitude 2 (d)none 14. Given vector , ˆ 3 ˆ A = 2i + j the angle between A and y-axis is (a) tan 3 / 2 −1 (b) tan 2 / 3 −1 (c) sin 2 / 3 −1 (d) cos 2 / 3 −1 15. The unit vector along i j ˆ + ˆ is (a) k ˆ (b) i j ˆ + ˆ (c) 2 ˆ ˆ i + j (d) 2 ˆ ˆ i + j 16. A vector is represented by i j k ˆ 2 ˆ ˆ 3 + + . Its length in XY plane is (a) 2 (b) 14 (c) 10 (d) 5 17. Five equal forces of 10 N each are applied at one point and all are lying in one plane. If the angles between them are equal, the resultant force will be (a) Zero (b) 10 N (c) 20 N (d) 10 2N 18. The angle made by the vector A i j ˆ ˆ = + with x- axis is (a) 90° (b) 45° (c) 22.5° (d) 30° 19. 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 20. Angular momentum is (a) A scalar (b) A polar vector (c) An axial vector (d) None of these 21. Which of the following is a vector (a) Pressure (b) Surface tension (c) Moment of inertia (d) None of these 22. If P Q   = then which of the following is NOT correct (a) P ˆ Q ˆ = (b) | P| | Q|   = (c) PQ QP ˆ ˆ = (d) P Q P ˆ Q ˆ + = +   23. The position vector of a particle is r a t i a t j ˆ ( sin ) ˆ = ( cos ) +   . The velocity of the particle is (a) Parallel to the position vector (b) Perpendicular to the position vector (c) Directed towards the origin (d) Directed away from the origin 24. Which of the following is a scalar quantity (a) Displacement (b) Electric field (c) Acceleration (d) Work 25. If a unit vector is represented by i j ck ˆ ˆ 0.8 ˆ 0.5 + + , then the value of ‘c’ is (a) 1 (b) 0.11 (c) 0.01 (d) 0.39 26. A boy walks uniformally along the sides of a rectangular park of size 400 m× 300 m, starting from one corner to the other corner diagonally opposite. Which of the following statement is incorrect (a) He has travelled a distance of 700 m (b) His displacement is 700 m (c) His displacement is 500 m (d) His velocity is not uniform throughout the walk 27. The unit vector parallel to the resultant of the vectors A i j k ˆ 6 ˆ 3 ˆ = 4 + +  and B i j k ˆ 8 ˆ 3 ˆ = − + −  is (a) ) ˆ 2 ˆ 6 ˆ (3 7 1 i + j − k (b) ) ˆ 2 ˆ 6 ˆ (3 7 1 i + j + k (c) ) ˆ 2 ˆ 6 ˆ (3 49 1 i + j − k (d) ) ˆ 2 ˆ 6 ˆ (3 49 1 i − j + k 28. Surface area is (a) Scalar (b)Vector (c) Neither scalar nor vector (d)Both scalar and vector 29. With respect to a rectangular cartesian coordinate system, three vectors are expressed as a i j ˆ ˆ = 4 −  , b i j ˆ 2 ˆ = −3 +  and c k ˆ = −  where i j k ˆ , ˆ , ˆ are unit vectors, along the X, Y and Z-axis respectively. The unit vectors r ˆ along the direction of sum of these vector is (a) ) ˆ ˆ ˆ ( 3 1 rˆ = i + j − k (b) ) ˆ ˆ ˆ ( 2 1 rˆ = i + j − k (c) ) ˆ ˆ ˆ ( 3 1 rˆ = i − j + k (d) ) ˆ ˆ ˆ ( 2 1 rˆ = i + j + k 30. The angle between the two vectors A i j k ˆ 5 ˆ 4 ˆ = 3 + +  and B i j k ˆ 5 ˆ 4 ˆ = 3 + +  is (a) 60° (b) Zero (c) 90° (d) None of these 31. The position vector of a particle is determined by the expression r t i t j k ˆ 7 ˆ 4 ˆ 3 2 2 = + +  The distance traversed in first 10 sec is (a) 500 m (b) 300 m (c) 150 m (d) 100 m 32. Unit vector parallel to the resultant of vectors A i j ˆ 3 ˆ = 4 −  and B i j ˆ 8 ˆ = 8 +  will be (a) 13 ˆ 5 ˆ 24 i + j (b) 13 ˆ 5 ˆ 12i + j (c) 13 ˆ 5 ˆ 6i + j (d) None of these 33. The component of vector A i j ˆ 3 ˆ = 2 + along the vector i j ˆ + ˆ is (a) 2 5 (b) 10 2 (c) 5 2 (d) 5 34. The angle between the two vectors A i j k ˆ 5 ˆ 4 ˆ = 3 + +  and B i j k ˆ 5 ˆ 4 ˆ = 3 + −  will be (a) 90° (b) 0° (c) 60° (d) 45°
35. There are two force vectors, one of 5 N and other of 12 N at what angle the two vectors be added to get resultant vector of 17 N, 7 N and 13 N respectively (a) 0°, 180° and 90° (b) 0°, 90° and 180° (c) 0°, 90° and 90° (d) 180°, 0° and 90° 36. If A i j ˆ 3 ˆ = 4 − and B i j ˆ 8 ˆ = 6 + then magnitude and direction of A + B will be (a) 5, tan (3 / 4) −1 (b) 5 5, tan (1 / 2) −1 (c) 10, tan (5) −1 (d) 25, tan (3 / 4) −1 37. A truck travelling due north at 20 m/s turns west and travels at the same speed. The change in its velocity be (a) 40 m/s N–W (b) 20 2 m/s N–W (c) 40 m/s S–W (d) 20 2 m/s S–W 38. If the sum of two unit vectors is a unit vector, then magnitude of difference is (a) 2 (b) 3 (c) 1 / 2 (d) 5 39. A i j B j k ˆ ˆ , 3 ˆ ˆ = 2 + = − and C i k ˆ 2 ˆ = 6 − , Value of A − 2B + 3C would be (a) i j k ˆ 4 ˆ 5 ˆ 20 + + (b) i j k ˆ 4 ˆ 5 ˆ 20 − − (c) i j k ˆ 20 ˆ 5 ˆ 4 + + (d) i j k ˆ 10 ˆ 4 ˆ 5 + + 40. An object of m kg with speed of v m/s strikes a wall at an angle  and rebounds at the same speed and same angle. The magnitude of the change in momentum of the object will be (a) 2m v cos (b) 2 m v sin (c) 0 (d) 2 m v 41. Two forces, each of magnitude F have a resultant of the same magnitude F. The angle between the two forces is (a) 45° (b) 120° (c) 150° (d) 60° 42. For the resultant of the two vectors to be maximum, what must be the angle between them (a) 0° (b) 60° (c) 90° (d) 180° 43. A particle is simultaneously acted by two forces equal to 4 N and 3 N. The net force on the particle is (a) 7 N (b) 5 N (c) 1 N (d) Between 1 N and 7 N 44. Two vectors 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 C  45. If the resultant of the two forces has a magnitude smaller than the magnitude of larger force, the two forces must be (a) Different both in magnitude and direction (b) Mutually perpendicular to one another (c) Possess extremely small magnitude (d) Point in opposite directions 46. 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) 2 2 2 F1 + F (d) 2 2 2 F1 + F 47. If | A − B| =| A| =| B|, the angle between A and B is (a) 60° (b) 0° (c) 120° (d) 90° 48. Let the angle between two nonzero vectors A and B be 120° 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| 49. 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 50. Magnitude of vector which comes on addition of two vectors, i j ˆ 7 ˆ 6 + and i j ˆ 4 ˆ 3 + is (a) 136 (b) 13.2 (c) 202 (d) 160 51. A particle has displacement of 12 m towards east and 5 m towards north then 6 m vertically upward. The sum of these displacements is (a) 12 (b) 10.04 m
(c) 14.31 m (d) None of these 52. The three vectors A i j k B i j k ˆ 5 ˆ 3 ˆ , ˆ ˆ 2 ˆ = 3 − + = − + and C i j k ˆ 4 ˆ ˆ = 2 + − form (a) An equilateral triangle (b) Isosceles triangle (c) A right angled triangle (d) No triangle 53. For the figure (a) A + B = C (b) B + C = A (c) C + A = B (d) A + B + C = 0 54. 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 55. The value of the sum of two vectors A and B with  as the angle between them is (a) 2 cos 2 2 A + B + AB (b) 2 cos 2 2 A − B + AB (c) 2 sin 2 2 A + B − AB (d) 2 sin 2 2 A + B + AB 56. 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, 23 (d) 10, 20, 40 57. When three forces of 50 N, 30 N and 15 N 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 58. The sum of two forces acting at a point is 16 N. If the resultant force is 8 N and its direction is perpendicular to minimum force then the forces are (a) 6 N and 10 N (b) 8 N and 8 N (c) 4 N and 12 N (d) 2 N and 14 N 59. If vectors P, Q and R have magnitude 5, 12 and 13 units and P + Q = R, the angle between Q and R is (a) 12 5 cos−1 (b) 13 5 cos−1 (c) 13 12 cos−1 (d) 13 7 cos−1 60. 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) 120° (b) 150° (c) 135° (d) None of these 61. What vector must be added to the two vectors i j k ˆ 2 ˆ 2 ˆ − + and , ˆ ˆ ˆ 2i + j − k so that the resultant may be a unit vector along x- axis (a) i j k ˆ ˆ ˆ 2 + − (b) i j k ˆ ˆ ˆ − 2 + − (c) i j k ˆ ˆ ˆ 2 − + (d) i j k ˆ ˆ ˆ − 2 − − 62. 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 − 63. 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 − 64. 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 65. 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) 66. 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 / 2) −1 (b) cos ( 1 / 2) 1 − − (c) cos ( 1 / 4) 1 − − (d) cos (1 / 4) −1 67. Given that A + B = C and that C is ⊥ to A . Further if | A| =| C|, then what is the angle between A and B (a) radian 4  (b) radian 2  (c) radian 4 3 (d)  radian 68. A body is at rest under the action of three forces, two of which are , ˆ , 6 ˆ 1 4 2 F = i F = j   the third force is (a) i j ˆ 6 ˆ 4 + (b) i j ˆ 6 ˆ 4 −