MCQs on Fundamentals for Vectors

Physics MCQs

1. The vector projection of a vector \(\mathbf{A} = 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}\) on y-axis is

(a) 5

(b) 4

(d) Zero

Correct Answer: (b) 4

The projection of a vector on the y-axis is simply its y-component. For \(\mathbf{A} = 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}\), the y-component is 4.

2. Position of a particle in a rectangular-co-ordinate system is (3, 2, 5). Then its position vector will be

(a) \(3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}\)

(b) \(3\mathbf{i} + 2\mathbf{j} - 5\mathbf{k}\)

(d) None of these

Correct Answer: (c) \(3\mathbf{i} + 2\mathbf{j} + 5\mathbf{k}\)

The position vector is formed by using the coordinates as components: \(x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\).

3. If a particle moves from point P (2,3,5) to point Q (3,4,5). Its displacement vector be

(a) \(\mathbf{i} + \mathbf{j} + 10\mathbf{k}\)

(b) \(\mathbf{i} + \mathbf{j}\)

(d) \(\mathbf{i} + \mathbf{j} + 5\mathbf{k}\)

Correct Answer: (b) \(\mathbf{i} + \mathbf{j}\)

Displacement vector = Final position - Initial position = \((3-2)\mathbf{i} + (4-3)\mathbf{j} + (5-5)\mathbf{k} = \mathbf{i} + \mathbf{j}\).

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

(d) 2.5 N

Correct Answer: (d) 2.5 N

Vertical component = \(F \cos \theta = 5 \cos 60° = 5 \times 0.5 = 2.5\) N.

5. If \(\mathbf{A} = \mathbf{i} + \mathbf{j} + \mathbf{k}\) and \(\mathbf{B} = 2\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\), the vector having the same magnitude as B and parallel to A is

(a) \(2\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\)

(b) \(\mathbf{i} + \mathbf{j} + \mathbf{k}\)

(d) \(\frac{2\sqrt{3}}{3}(\mathbf{i} + \mathbf{j} + \mathbf{k})\)

Correct Answer: (c) \(\sqrt{3}(\mathbf{i} + \mathbf{j} + \mathbf{k})\)

Magnitude of B is \(2\sqrt{3}\). Unit vector in direction of A is \(\frac{\mathbf{i} + \mathbf{j} + \mathbf{k}}{\sqrt{3}}\). Multiply by magnitude of B to get the required vector.

6. Vector \(\mathbf{A}\) makes equal angles with x, y and z axis. Value of its components (in terms of magnitude of \(\mathbf{A}\)) will be

(a) \(\frac{A}{\sqrt{2}}\)

(b) \(\frac{A}{\sqrt{3}}\)

(d) \(\frac{A}{2}\)

Correct Answer: (b) \(\frac{A}{\sqrt{3}}\)

For equal angles, \(l = m = n\) and \(l^2 + m^2 + n^2 = 1\). Thus each component is \(\frac{1}{\sqrt{3}}\) times the magnitude.

7. If \(\mathbf{A} = 2\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}\), the direction of cosines of the vector \(\mathbf{A}\) are

(a) \(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\)

(b) \(\frac{2}{5}, \frac{3}{5}, \frac{6}{5}\)

(d) \(\frac{2}{49}, \frac{3}{49}, \frac{6}{49}\)

Correct Answer: (a) \(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\)

Direction cosines are components divided by magnitude. Magnitude = \(\sqrt{4+9+36} = 7\). Thus direction cosines are \(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\).

8. The vector that must be added to the vector \(\mathbf{i} - \mathbf{j}\) and \(2\mathbf{i} + 3\mathbf{j}\) so that the resultant vector is a unit vector along the y-axis is

(a) \(-3\mathbf{i} - \mathbf{j}\)

(b) \(3\mathbf{i} - \mathbf{j}\)

(d) Null vector

Correct Answer: (a) \(-3\mathbf{i} - \mathbf{j}\)

Let the vector to be added be \(\mathbf{V}\). Then \((\mathbf{i} - \mathbf{j}) + (2\mathbf{i} + 3\mathbf{j}) + \mathbf{V} = \mathbf{j}\). Solving gives \(\mathbf{V} = -3\mathbf{i} - \mathbf{j}\).

9. How many minimum number of coplanar vectors having different magnitudes can be added to give zero resultant

(a) 2

(b) 3

(d) 5

Correct Answer: (b) 3

The minimum number of coplanar vectors needed to form a closed polygon (zero resultant) with different magnitudes is 3.

10. A hall has the dimensions 10m × 12m × 14m. 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

(d) 20 m

Correct Answer: (d) 20 m

Displacement is the diagonal of the rectangular prism: \(\sqrt{10^2 + 12^2 + 14^2} = \sqrt{100 + 144 + 196} = \sqrt{440} \approx 20\) m.

Physics MCQs

11. 100 coplanar forces each equal to 10 N act on a body. Each force makes angle \(\frac{2\pi}{100}\) with the preceding force. What is the resultant of the forces

(a) 1000 N

(b) 500 N

(d) Zero

Correct Answer: (d) Zero

When 100 equal forces are arranged with equal angular separation in a plane, they form a closed polygon. The vector sum (resultant) of forces in a closed polygon is zero.

12. The magnitude of a given vector with end points (4, -4, 0) and (-2, -2, 0) must be

(a) 6

(b) \(5\sqrt{2}\)

(d) \(2\sqrt{10}\)

Correct Answer: (d) \(2\sqrt{10}\)

The vector components are: \((-2-4, -2-(-4), 0-0) = (-6, 2, 0)\).
Magnitude = \(\sqrt{(-6)^2 + 2^2 + 0^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}\).

13. The expression \(\frac{1}{\sqrt{2}}(\hat{i} + \hat{j})\) is a

(a) Unit vector

(b) Null vector

(d) Scalar

Correct Answer: (a) Unit vector

Magnitude = \(\sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1\).
Since the magnitude is 1, it's a unit vector.

14. Given vector \(\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}\), the angle between \(\vec{A}\) and y-axis is

(a) \(\sin^{-1}\left(\frac{2}{\sqrt{29}}\right)\)

(b) \(\cos^{-1}\left(\frac{3}{\sqrt{29}}\right)\)

(d) \(\cot^{-1}\left(\frac{4}{3}\right)\)

Correct Answer: (b) \(\cos^{-1}\left(\frac{3}{\sqrt{29}}\right)\)

The angle θ with y-axis is given by \(\cosθ = \frac{A_y}{|\vec{A}|} = \frac{3}{\sqrt{2^2 + 3^2 + 4^2}} = \frac{3}{\sqrt{29}}\).

15. The unit vector along \(\hat{i} + \hat{j}\) is

(a) \(\hat{i} + \hat{j}\)

(b) \(\hat{i} + \hat{j}\)

(d) \(\frac{1}{2}(\hat{i} + \hat{j})\)

Correct Answer: (c) \(\frac{1}{\sqrt{2}}(\hat{i} + \hat{j})\)

Unit vector = \(\frac{\vec{A}}{|\vec{A}|} = \frac{\hat{i} + \hat{j}}{\sqrt{1^2 + 1^2}} = \frac{1}{\sqrt{2}}(\hat{i} + \hat{j})\).

16. A vector is represented by \(3\hat{i} + 2\hat{j} + \hat{k}\). Its length in XY plane is

(a) 2

(b) \(\sqrt{10}\)

(d) \(\sqrt{14}\)

Correct Answer: (c) \(\sqrt{13}\)

In XY plane, we ignore the z-component. Length = \(\sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}\).

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

(d) \(10\sqrt{2}\) N

Correct Answer: (a) Zero

Five equal forces with equal angular separation (72° between each) in a plane will form a closed pentagon. The vector sum of forces in a closed polygon is zero.

18. The angle made by the vector \(\hat{i} + \hat{j}\) with x-axis is

(a) 90°

(b) 45°

(d) 30°

Correct Answer: (b) 45°

Angle with x-axis θ is given by \(\tanθ = \frac{A_y}{A_x} = \frac{1}{1} = 1\). Thus, θ = 45°.

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

(d) It is not possible to resolve a vector

Correct Answer: (b) Mutually perpendicular vectors which have the original vector as their resultant

Any vector can be resolved into components along mutually perpendicular axes (like x, y, z components in 3D space).

20. Angular momentum is

(a) A scalar

(b) A polar vector

(d) None of these

Correct Answer: (c) An axial vector

Angular momentum is an axial vector (or pseudovector) because it involves the cross product of position and momentum vectors, and its direction is given by the right-hand rule.

Physics MCQs

20. Angular momentum is

(a) A scalar

(b) A polar vector

(d) None of these

Correct Answer: (c) An axial vector

Angular momentum is an axial vector (also called a pseudovector) because its direction depends on the handedness of the coordinate system. Unlike polar vectors, axial vectors don't reverse direction when the coordinate axes are inverted.

21. Which of the following is a vector

(a) Pressure

(b) Surface tension

(d) None of these

Correct Answer: (d) None of these

All the given options are scalar quantities:

Pressure is a scalar (force per unit area)
Surface tension is a scalar (force per unit length)
Moment of inertia is a scalar (rotational analog of mass)

24. Which of the following is a scalar quantity

(a) Displacement

(b) Electric field

(d) Work

Correct Answer: (d) Work

Work is a scalar quantity (dot product of force and displacement). The other options are all vectors:

Displacement is a vector
Electric field is a vector
Acceleration is a vector

25. If a unit vector is represented by \(0.5\hat{i} + 0.8\hat{j} + c\hat{k}\), then the value of 'c' is

(a) 1

(b) \(\sqrt{0.8}\)

(d) \(\sqrt{0.01}\)

Correct Answer: (c) \(\sqrt{0.11}\)

For a unit vector, the magnitude must be 1: \[ \sqrt{(0.5)^2 + (0.8)^2 + c^2} = 1 \\ \sqrt{0.25 + 0.64 + c^2} = 1 \\ \sqrt{0.89 + c^2} = 1 \\ 0.89 + c^2 = 1 \\ c^2 = 1 - 0.89 = 0.11 \\ c = \sqrt{0.11} \]

26. A boy walks uniformly 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

(d) His velocity is not uniform throughout the walk

Correct Answer: (b) His displacement is 700 m

Distance traveled = 400 m + 300 m = 700 m (correct)
Displacement = diagonal length = \(\sqrt{400^2 + 300^2} = 500\) m (so option b is incorrect and c is correct)
Velocity is not uniform because direction changes when turning corners (correct)

27. The unit vector parallel to the resultant of the vectors \(2\hat{i} + 3\hat{j} - \hat{k}\) and \(\hat{i} - 2\hat{j} + \hat{k}\) is

(a) \(\frac{3\hat{i} + \hat{j}}{\sqrt{10}}\)

(b) \(\frac{3\hat{i} + \hat{j}}{\sqrt{8}}\)

(d) \(\frac{3\hat{i} - \hat{j}}{\sqrt{8}}\)

Correct Answer: (a) \(\frac{3\hat{i} + \hat{j}}{\sqrt{10}}\)

First find the resultant vector: \[ \vec{R} = (2\hat{i} + 3\hat{j} - \hat{k}) + (\hat{i} - 2\hat{j} + \hat{k}) = 3\hat{i} + \hat{j} \] Then find the unit vector: \[ \text{Magnitude} = \sqrt{3^2 + 1^2} = \sqrt{10} \\ \text{Unit vector} = \frac{3\hat{i} + \hat{j}}{\sqrt{10}} \]

28. Surface area is

(a) Scalar

(b) Vector

(d) Both scalar and vector

Correct Answer: (a) Scalar

Surface area is a scalar quantity as it only has magnitude and no direction. It's the measure of the total area that the surface of an object occupies.

29. With respect to a rectangular cartesian coordinate system, three vectors are expressed as \(\vec{a} = 4\hat{i} - \hat{j}\), \(\vec{b} = -3\hat{i} + 2\hat{j}\) and \(\vec{c} = -\hat{k}\) where \(\hat{i}, \hat{j}, \hat{k}\) are unit vectors, along the X, Y and Z-axis respectively. The unit vector along the direction of sum of these vector is

(a) \(\frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{3}}\)

(b) \(\frac{\hat{i} - \hat{j} + \hat{k}}{\sqrt{3}}\)

(d) \(\frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{2}}\)

Correct Answer: (a) \(\frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{3}}\)

First find the sum of vectors: \[ \vec{a} + \vec{b} + \vec{c} = (4\hat{i} - \hat{j}) + (-3\hat{i} + 2\hat{j}) + (-\hat{k}) = \hat{i} + \hat{j} - \hat{k} \] Then find the unit vector: \[ \text{Magnitude} = \sqrt{1^2 + 1^2 + (-1)^2} = \sqrt{3} \\ \text{Unit vector} = \frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{3}} \]

30. The angle between the two vectors \(\vec{A} = 3\hat{i} + 4\hat{j}\) and \(\vec{B} = -3\hat{i} + 4\hat{j}\) is

(a) 60°

(b) Zero

(d) None of these

Correct Answer: (d) None of these

To find the angle θ between vectors: \[ \vec{A} \cdot \vec{B} = (3)(-3) + (4)(4) = -9 + 16 = 7 \\ |\vec{A}| = \sqrt{3^2 + 4^2} = 5 \\ |\vec{B}| = \sqrt{(-3)^2 + 4^2} = 5 \\ \cosθ = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}||\vec{B}|} = \frac{7}{25} \\ θ = \cos^{-1}\left(\frac{7}{25}\right) \approx 73.74° \] This angle is not any of the given options (60°, 0°, or 90°).