Vectors Problems

This section provides 100 problems to test your understanding of vectors, including representation, properties, addition, subtraction, components, unit vectors, dot and cross products, and their applications in physics. Inspired by JEE Main, JEE Advanced, and NEET exam patterns, these problems are tailored for exam preparation, offering a mix of numerical, conceptual, and derivation-based challenges. NEET-style problems (66–100) are formatted as multiple-choice questions (MCQs) to match the exam’s objective format. Problems are organized by type to support progressive learning and build confidence in mastering vectors, a fundamental concept for JEE/NEET success.

Numerical Problems

1. A vector $\vec{A}$ has components $A_x = 8$ and $A_y = 6$. Calculate its magnitude.

   • (a) $10$
   • (b) $12$
   • (c) $14$
   • (d) $16$

2. A vector $\vec{B}$ has components $B_x = 5$ and $B_y = -12$. Find the angle it makes with the x-axis.

   • (a) $-67.38^\circ$
   • (b) $67.38^\circ$
   • (c) $22.62^\circ$
   • (d) $-22.62^\circ$

3. Two vectors $\vec{A} = 3 \hat{i} + 4 \hat{j}$ and $\vec{B} = 1 \hat{i} - 2 \hat{j}$ are added. Calculate the magnitude of the resultant.

   • (a) $5$
   • (b) $6$
   • (c) $7$
   • (d) $8$

4. A vector $\vec{C}$ has magnitude $10$ and makes an angle of $30^\circ$ with the x-axis. Find its x-component.

   • (a) $5$
   • (b) $8.66$
   • (c) $10$
   • (d) $12$

5. A vector $\vec{D} = 2 \hat{i} + 3 \hat{j}$ is scaled by a factor of $3$. Calculate the magnitude of the new vector.

   • (a) $5$
   • (b) $10$
   • (c) $15$
   • (d) $20$

6. Find the dot product of $\vec{A} = 4 \hat{i} + 3 \hat{j}$ and $\vec{B} = 2 \hat{i} - 1 \hat{j}$.

   • (a) $5$
   • (b) $6$
   • (c) $7$
   • (d) $8$

7. Calculate the magnitude of the cross product of $\vec{A} = 5 \hat{i} + 2 \hat{j}$ and $\vec{B} = 1 \hat{i} + 3 \hat{j}$.

   • (a) $11$
   • (b) $12$
   • (c) $13$
   • (d) $14$

8. A vector $\vec{E}$ in 3D is given as $\vec{E} = 2 \hat{i} + 3 \hat{j} - 6 \hat{k}$. Find its magnitude.

   • (a) $5$
   • (b) $7$
   • (c) $9$
   • (d) $11$

9. A vector $\vec{F}$ has magnitude $15$ at $60^\circ$ to the x-axis. Find its y-component.

   • (a) $7.5$
   • (b) $10$
   • (c) $12.99$
   • (d) $15$

10. The resultant of $\vec{A} = 1 \hat{i} + 2 \hat{j}$ and $\vec{B} = -2 \hat{i} + 3 \hat{j}$ is subtracted from $\vec{C} = 4 \hat{i} - 1 \hat{j}$. Find the magnitude of the final vector.

    • (a) $5$
    • (b) $6$
    • (c) $7$
    • (d) $8$

11. A unit vector is derived from $\vec{A} = 3 \hat{i} + 4 \hat{j}$. Find the x-component of the unit vector.

    • (a) $0.6$
    • (b) $0.8$
    • (c) $0.4$
    • (d) $0.2$

12. Two vectors $\vec{A} = 6 \hat{i} + 8 \hat{j}$ and $\vec{B} = -2 \hat{i} + 3 \hat{j}$ are added. Find the angle of the resultant with the x-axis.

    • (a) $36.87^\circ$
    • (b) $53.13^\circ$
    • (c) $63.43^\circ$
    • (d) $75.96^\circ$

13. A force $\vec{F} = 5 \hat{i} + 5 \hat{j} , N$ acts over a displacement $\vec{d} = 2 \hat{i} + 1 \hat{j} , m$. Calculate the work done.

    • (a) $10 , J$
    • (b) $15 , J$
    • (c) $20 , J$
    • (d) $25 , J$

14. A torque is produced by $\vec{F} = 3 \hat{i} + 2 \hat{j} , N$ at $\vec{r} = 1 \hat{i} - 4 \hat{j} , m$. Find the magnitude of the torque.

    • (a) $10 , N \cdot m$
    • (b) $12 , N \cdot m$
    • (c) $14 , N \cdot m$
    • (d) $16 , N \cdot m$

15. A vector $\vec{A}$ has magnitude $20$ and makes $45^\circ$ with the x-axis. Find the magnitude of its projection on the y-axis.

    • (a) $10$
    • (b) $14.14$
    • (c) $17.32$
    • (d) $20$

16. Add vectors $\vec{A} = 4 \hat{i} - 3 \hat{j}$ and $\vec{B} = -4 \hat{i} + 3 \hat{j}$. Find the magnitude of the resultant.

    • (a) $0$
    • (b) $1$
    • (c) $2$
    • (d) $3$

17. A 3D vector $\vec{C} = 1 \hat{i} - 2 \hat{j} + 2 \hat{k}$ is scaled by $-2$. Find the magnitude of the new vector.

    • (a) $3$
    • (b) $6$
    • (c) $9$
    • (d) $12$

18. Find the dot product of $\vec{A} = 2 \hat{i} + 2 \hat{j} + 2 \hat{k}$ and $\vec{B} = 1 \hat{i} - 1 \hat{j} + 1 \hat{k}$.

    • (a) $1$
    • (b) $2$
    • (c) $3$
    • (d) $4$

19. Calculate the magnitude of the cross product of $\vec{A} = 7 \hat{i} + 1 \hat{j}$ and $\vec{B} = 2 \hat{i} + 5 \hat{j}$.

    • (a) $33$
    • (b) $34$
    • (c) $35$
    • (d) $36$

20. A vector $\vec{D}$ has components $D_x = -3$, $D_y = 4$, $D_z = 12$. Find its magnitude.

    • (a) $13$
    • (b) $14$
    • (c) $15$
    • (d) $16$

21. A vector $\vec{E}$ of magnitude $8$ makes $60^\circ$ with the x-axis. Find its x-component.

    • (a) $4$
    • (b) $6$
    • (c) $6.93$
    • (d) $8$

22. Subtract $\vec{B} = 2 \hat{i} - 3 \hat{j}$ from $\vec{A} = 5 \hat{i} + 1 \hat{j}$. Find the magnitude of the result.

    • (a) $5$
    • (b) $6$
    • (c) $7$
    • (d) $8$

23. A unit vector is derived from $\vec{A} = 1 \hat{i} + 1 \hat{j} + 1 \hat{k}$. Find the z-component of the unit vector.

    • (a) $0.333$
    • (b) $0.577$
    • (c) $0.667$
    • (d) $0.866$

24. Two vectors $\vec{A} = 9 \hat{i} + 12 \hat{j}$ and $\vec{B} = -3 \hat{i} + 4 \hat{j}$ are added. Find the magnitude of the resultant.

    • (a) $10$
    • (b) $15$
    • (c) $20$
    • (d) $25$

25. A force $\vec{F} = 4 \hat{i} - 3 \hat{j} , N$ acts over $\vec{d} = 1 \hat{i} + 2 \hat{j} , m$. Calculate the work done.

    • (a) $-2 , J$
    • (b) $0 , J$
    • (c) $2 , J$
    • (d) $4 , J$

26. A torque is produced by $\vec{F} = 2 \hat{i} + 5 \hat{j} , N$ at $\vec{r} = 3 \hat{i} - 1 \hat{j} , m$. Find the magnitude of the torque.

    • (a) $17 , N \cdot m$
    • (b) $18 , N \cdot m$
    • (c) $19 , N \cdot m$
    • (d) $20 , N \cdot m$

27. A vector $\vec{A}$ has magnitude $25$ at $53^\circ$ to the x-axis. Find the magnitude of its projection on the x-axis.

    • (a) $15$
    • (b) $20$
    • (c) $25$
    • (d) $30$

28. Add vectors $\vec{A} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k}$ and $\vec{B} = -3 \hat{i} - 4 \hat{j} - 5 \hat{k}$. Find the magnitude of the resultant.

    • (a) $0$
    • (b) $1$
    • (c) $2$
    • (d) $3$

29. A vector $\vec{A} = 6 \hat{i} - 8 \hat{j}$ is scaled by $0.5$. Find the magnitude of the new vector.

    • (a) $5$
    • (b) $10$
    • (c) $15$
    • (d) $20$

30. Find the dot product of $\vec{A} = 1 \hat{i} + 2 \hat{j} - 3 \hat{k}$ and $\vec{B} = 2 \hat{i} - 1 \hat{j} + 1 \hat{k}$.

    • (a) $-3$
    • (b) $-2$
    • (c) $-1$
    • (d) $0$

31. Calculate the magnitude of the cross product of $\vec{A} = 4 \hat{i} + 4 \hat{j}$ and $\vec{B} = 3 \hat{i} + 3 \hat{j}$.

    • (a) $0$
    • (b) $1$
    • (c) $2$
    • (d) $3$

32. A vector $\vec{A}$ has components $A_x = 2$, $A_y = -2$, $A_z = 1$. Find its magnitude.

    • (a) $3$
    • (b) $4$
    • (c) $5$
    • (d) $6$

33. A vector $\vec{B}$ of magnitude $12$ makes $45^\circ$ with the x-axis. Find its y-component.

    • (a) $6$
    • (b) $8.49$
    • (c) $10$
    • (d) $12$

34. Subtract $\vec{B} = 1 \hat{i} + 2 \hat{j} + 3 \hat{k}$ from $\vec{A} = 4 \hat{i} + 5 \hat{j} + 6 \hat{k}$. Find the magnitude of the result.

    • (a) $3$
    • (b) $4$
    • (c) $5$
    • (d) $6$

35. A unit vector is derived from $\vec{A} = 2 \hat{i} - 2 \hat{j}$. Find the y-component of the unit vector.

    • (a) $-0.707$
    • (b) $-0.5$
    • (c) $0.5$
    • (d) $0.707$

Conceptual Problems

36. What is the main difference between a vector and a scalar?

• (a) Vector has magnitude only, scalar has direction
• (b) Vector has magnitude and direction, scalar has magnitude only
• (c) Both have magnitude and direction
• (d) Both have magnitude only

37. What does a negative vector indicate?

• (a) Zero magnitude
• (b) Opposite direction
• (c) Same direction
• (d) Perpendicular direction

38. Which property holds for vector addition?

• (a) Commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$
• (b) Non-commutative: $\vec{A} + \vec{B} \neq \vec{B} + \vec{A}$
• (c) Only associative
• (d) Only distributive

39. What is the result of the dot product of two perpendicular vectors?

• (a) Zero
• (b) One
• (c) Negative one
• (d) Depends on their magnitudes

40. What is the direction of the cross product $\vec{A} \times \vec{B}$?

• (a) Parallel to $\vec{A}$
• (b) Parallel to $\vec{B}$
• (c) Perpendicular to the plane of $\vec{A}$ and $\vec{B}$
• (d) Opposite to $\vec{A}$

41. What is the magnitude of a unit vector?

• (a) $0$
• (b) $1$
• (c) Depends on the vector
• (d) Infinite

42. What does the dot product $\vec{A} \cdot \vec{B}$ represent physically?

• (a) Torque
• (b) Work done by $\vec{A}$ along $\vec{B}$
• (c) Angular momentum
• (d) Area of parallelogram

43. What is the result of the cross product of two parallel vectors?

• (a) Zero vector
• (b) Unit vector
• (c) Scalar
• (d) Non-zero vector

44. How are vectors equal?

• (a) Same magnitude only
• (b) Same direction only
• (c) Same magnitude and direction
• (d) Same starting point

45. What is the dimension of a position vector?

• (a) $[\text{L}]$
• (b) $[\text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}]$

46. What does a zero vector imply?

• (a) No magnitude
• (b) No direction
• (c) No magnitude and direction
• (d) Perpendicular direction

47. Which property does not hold for cross products?

• (a) Commutative: $\vec{A} \times \vec{B} = \vec{B} \times \vec{A}$
• (b) Associative: $(\vec{A} \times \vec{B}) \times \vec{C} = \vec{A} \times (\vec{B} \times \vec{C})$
• (c) Distributive: $\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$
• (d) All hold

48. What is the physical significance of the cross product $\vec{r} \times \vec{F}$?

• (a) Work
• (b) Torque
• (c) Displacement
• (d) Velocity

49. What is the magnitude of $\hat{i} \cdot \hat{j}$?

• (a) $0$
• (b) $1$
• (c) $-1$
• (d) $2$

50. How does scalar multiplication affect a vector’s direction?

• (a) Always changes direction
• (b) Reverses if scalar is negative, same if positive
• (c) Never changes direction
• (d) Depends on the vector’s magnitude

Derivation Problems

51. Derive the magnitude of a vector $\vec{A}$ with components $A_x$, $A_y$, $A_z$ in 3D.

52. Derive the expression for the resultant of two vectors $\vec{A}$ and $\vec{B}$ using the component method.

53. Derive the dot product $\vec{A} \cdot \vec{B}$ in terms of components for 2D vectors.

54. Derive the cross product $\vec{A} \times \vec{B}$ in terms of components for 2D vectors.

55. Derive the unit vector of a given vector $\vec{A}$ in 3D.

56. Derive the magnitude of the resultant of two vectors using the parallelogram law.

57. Derive the expression for work done $W = \vec{F} \cdot \vec{d}$ using components.

58. Derive the torque $\vec{\tau} = \vec{r} \times \vec{F}$ in component form for 2D vectors.

59. Derive the angle between two vectors $\vec{A}$ and $\vec{B}$ using the dot product.

60. Derive the magnitude of a vector after scalar multiplication by a factor $k$.

61. Derive the projection of $\vec{A}$ onto $\vec{B}$ using the dot product.

62. Derive the position vector of a point in 3D space relative to the origin.

63. Derive the displacement vector between two points in 2D space.

64. Derive the commutative property of vector addition using components.

65. Derive the distributive property $k (\vec{A} + \vec{B}) = k \vec{A} + k \vec{B}$ for scalar multiplication.

NEET-style Conceptual Problems

66. What is the unit of a displacement vector in SI units?

• (a) $m/s$
• (b) $m$
• (c) $m/s^2$
• (d) $s$

67. What does a zero dot product between two vectors indicate?

• (a) Parallel vectors
• (b) Perpendicular vectors
• (c) Opposite vectors
• (d) Equal vectors

68. Which of the following is a vector quantity?

• (a) Speed
• (b) Distance
• (c) Velocity
• (d) Time

69. What is the magnitude of the cross product of two parallel vectors?

• (a) Zero
• (b) One
• (c) Depends on their magnitudes
• (d) Negative one

70. What is the dimension of a force vector?

• (a) $[\text{M} \text{L} \text{T}^{-2}]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-2}]$

71. What does the unit vector $\hat{A}$ represent?

• (a) Magnitude of $\vec{A}$
• (b) Direction of $\vec{A}$
• (c) Opposite direction of $\vec{A}$
• (d) Perpendicular to $\vec{A}$

72. What is the role of the dot product in calculating work?

• (a) Gives the magnitude of the force
• (b) Gives the component of force along displacement
• (c) Gives the torque
• (d) Gives the angular momentum

73. What happens to the cross product $\vec{A} \times \vec{B}$ if $\vec{A}$ and $\vec{B}$ are perpendicular?

• (a) Zero vector
• (b) Maximum magnitude
• (c) Unit vector
• (d) Scalar quantity

74. Why are position vectors useful in physics?

• (a) To calculate speed
• (b) To describe location relative to an origin
• (c) To calculate work
• (d) To calculate torque

75. What is the unit of torque in SI units?

• (a) $N$
• (b) $N \cdot m$
• (c) $J$
• (d) $m/s$

76. What does a negative scalar multiplication do to a vector?

• (a) Increases its magnitude
• (b) Reverses its direction
• (c) Makes it perpendicular
• (d) Reduces its magnitude to zero

77. Which kinematic equation does not involve a vector directly?

• (a) $v = u + at$
• (b) $x = ut + \frac{1}{2} at^2$
• (c) $v^2 = u^2 + 2ax$
• (d) $x = \frac{1}{2}(u + v)t$

78. What is the direction of the cross product $\vec{A} \times \vec{B}$ if $\vec{A}$ and $\vec{B}$ lie in the xy-plane?

• (a) Along x-axis
• (b) Along y-axis
• (c) Along z-axis
• (d) In the xy-plane

79. What is the physical significance of $\vec{F} \cdot \vec{d}$?

• (a) Torque
• (b) Work
• (c) Angular momentum
• (d) Displacement

80. What does a zero vector indicate?

• (a) No magnitude and direction
• (b) Perpendicular direction
• (c) Parallel direction
• (d) Opposite direction

81. What is the dimension of angular momentum, which involves a cross product?

• (a) $[\text{M} \text{L}^2 \text{T}^{-2}]$
• (b) $[\text{M} \text{L}^2 \text{T}^{-1}]$
• (c) $[\text{M} \text{L} \text{T}^{-2}]$
• (d) $[\text{L}^2 \text{T}^{-1}]$

82. What is the role of the cross product in calculating torque?

• (a) Gives the component of force along displacement
• (b) Gives the perpendicular component of force
• (c) Gives the work done
• (d) Gives the velocity

83. What happens to the dot product $\vec{A} \cdot \vec{B}$ if $\vec{A}$ and $\vec{B}$ are parallel?

• (a) Zero
• (b) Maximum magnitude
• (c) Negative magnitude
• (d) Unit vector

84. Which quantity is a scalar in vector operations?

• (a) Displacement
• (b) Velocity
• (c) Dot product
• (d) Cross product

85. What does a negative dot product indicate?

• (a) Vectors are perpendicular
• (b) Vectors are parallel
• (c) Angle between vectors is greater than $90^\circ$
• (d) Angle between vectors is less than $90^\circ$

86. What is the unit of work in SI units?

• (a) $N$
• (b) $N \cdot m$
• (c) $m/s$
• (d) $kg$

87. What does the cross product $\vec{v} \times \vec{B}$ represent in electromagnetism?

• (a) Electric force
• (b) Magnetic force direction
• (c) Work done
• (d) Torque

88. Why are unit vectors useful?

• (a) To increase magnitude
• (b) To specify direction
• (c) To calculate work
• (d) To calculate torque

89. What is the magnitude of $\hat{i} \times \hat{j}$?

• (a) $0$
• (b) $1$
• (c) $-1$
• (d) $2$

90. How does vector subtraction $\vec{A} - \vec{B}$ relate to vector addition?

• (a) $\vec{A} - \vec{B} = \vec{A} + \vec{B}$
• (b) $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$
• (c) $\vec{A} - \vec{B} = -\vec{A} + \vec{B}$
• (d) $\vec{A} - \vec{B} = \vec{B} - \vec{A}$

91. What is the role of the dot product in finding the angle between vectors?

• (a) Gives the magnitude
• (b) Gives the cosine of the angle
• (c) Gives the sine of the angle
• (d) Gives the direction

92. What does a zero cross product between two vectors indicate?

• (a) Perpendicular vectors
• (b) Parallel or anti-parallel vectors
• (c) Equal vectors
• (d) Opposite vectors

93. What is the physical significance of $\vec{r} \times \vec{p}$?

• (a) Work
• (b) Torque
• (c) Angular momentum
• (d) Velocity

94. What is the dimension of work, which involves a dot product?

• (a) $[\text{M} \text{L}^2 \text{T}^{-2}]$
• (b) $[\text{M} \text{L} \text{T}^{-2}]$
• (c) $[\text{L}^2 \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

95. Why is the cross product a vector quantity?

• (a) It has magnitude only
• (b) It has direction only
• (c) It has both magnitude and direction
• (d) It has neither magnitude nor direction

NEET-style Numerical Problems

96. A force $\vec{F} = 3 \hat{i} + 4 \hat{j} , N$ acts over $\vec{d} = 1 \hat{i} + 1 \hat{j} , m$. What is the work done?

• (a) $5 , J$
• (b) $6 , J$
• (c) $7 , J$
• (d) $8 , J$

97. A ball is projected with velocity $\vec{v} = 10 \hat{i} + 10 \hat{j} , m/s$. What is the magnitude of the velocity?

• (a) $10 , m/s$
• (b) $14.14 , m/s$
• (c) $17.32 , m/s$
• (d) $20 , m/s$

98. A particle experiences forces $\vec{F}_1 = 2 \hat{i} + 3 \hat{j} , N$ and $\vec{F}_2 = -1 \hat{i} + 5 \hat{j} , N$. What is the magnitude of the resultant force?

• (a) $5 , N$
• (b) $6 , N$
• (c) $7 , N$
• (d) $8 , N$

99. A torque is produced by $\vec{F} = 5 \hat{i} + 2 \hat{j} , N$ at $\vec{r} = 1 \hat{i} - 3 \hat{j} , m$. What is the magnitude of the torque?

• (a) $17 , N \cdot m$
• (b) $18 , N \cdot m$
• (c) $19 , N \cdot m$
• (d) $20 , N \cdot m$

100. A vector $\vec{A}$ has magnitude $6$ at $30^\circ$ to the x-axis. What is the magnitude of its x-component?
     - (a) $3$
     - (b) $5.20$
     - (c) $6$
     - (d) $7$

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