## Vector Multiple Choice Questions And Answers

**Question 1. What is the condition for \(\vec{A}+\vec{B}=\vec{A}-\vec{B}\) to be valid?**

- \(\vec{A}=0\)
- \(\vec{B}=0\)
- \(\vec{A}=\vec{B}\)
- \(\vec{A}=-\vec{B}\)

**Answer:** 2. \(\vec{B}=0\)

**Question 2. The magnitudes of the sum and difference of the two vectors are equal. The angle between the vectors is**

- 0°
- 90°
- 120°
- 60°

**Answer:** 2. 90°

**Question 3. The magnitude of each of the two vectors is P. The Magnitude of the resultant of the two is also P. The Angle between the vectors is**

- 0°
- 60°
- 120°
- 90°

**Answer:** 3. 120°

**Question 4.Two forces of equal magnitude act simultaneously on a particle. If the resultant of the forces is equal to the magnitude of each of them then the angle between the forces is**

- An acute angle
- An obtuse angle
- A right angle
- Of any value

**Answer:** 2. An obtuse angle

**Question 5. A man travels 30 m to the north, then 20 m to the east, and after that 30√2 m to the south-west. His displacement from the starting point is**

- 15 m to the east
- 28 m to the south
- 10 m to the west
- 15 m to the southwest

**Answer:** 3. 10 m to the west

**Question 6. \(0.2 \hat{i}+0.6 \hat{j}+a \hat{k}\) is a unit vector. The value of a should be**

- 0.3
- 0.4
- 0.6
- 0.8

**Answer:** 3. 0.6

**Question 7. \(\vec{R}\) is the resultant of the vectors \(\vec{A}\) and \(\vec{B}\). If R = \(\frac{B}{\sqrt{2}}\), then the angle θ is**

- 30°
- 45°
- 60°
- 75°

**Answer:** 2. 45°

**Question 8. The position vector of a particle is related to time t as \(\vec{r}=\left(t^2-1\right) \hat{i}+2 t \hat{j}\). The locus of the particle on the x-y plane is**

- Parabolic
- Circular
- Straight line
- Elliptical

**Answer:** 1. Parabolic

**Question 9. Two forces of the same magnitude F are at right angles to each other. The magnitude of the net force (total force) acting on the object is**

- F
- 2 F
- Between F and 2F
- More than 2 F

**Answer:** 3. Between F and 2 F

**Question 10. If a and b are two unit vectors inclined at an angle of 60° to each other, then**

- \(|a+b|>1\)
- \(|a+b|<1\)
- \(|a-b|>1\)
- \(|a-b|<1\)

**Answer:** 1. \(|a+b|>1\)

**Question 11. The condition (a-b)² = a²b² is satisfied when**

- a is parallel to b
- a ≠ b
- a-b = 1
- a ⊥ b

**Answer:** 3. a-b = 1

**Question 12. If \(\vec{P}+\vec{Q}=\vec{R} \text { and }|\vec{P}|=|\vec{Q}|=\sqrt{3} \text { and }|\vec{R}|=3\), then the angle between \(\vec{P}\) and \(\vec{Q}\) is**

- \(\frac{\pi}{4}\)
- \(\frac{\pi}{6}\)
- \(\frac{\pi}{3}\)
- \(\frac{\pi}{2}\)

**Answer:** 3. \(\frac{\pi}{3}\)

**Question 13. The angles which the vector \(\vec{A}=3 \hat{i}+6 \hat{j}+2 \hat{k}\) makes with the coordinate axes are**

- \(\cos ^{-1} \frac{3}{7}, \cos ^{-1} \frac{6}{7}\) and \(\cos ^{-1} \frac{2}{7}\)
- \(\cos ^{-1} \frac{4}{7}, \cos ^{-1} \frac{5}{7} \)and \(\cos ^{-1} \frac{3}{7}\)
- \(\cos ^{-1} \frac{3}{7}, \cos ^{-1} \frac{4}{7}\) and \(\cos ^{-1} \frac{1}{7}\)
- None of the above

**Answer:** 1. \(\cos ^{-1} \frac{3}{7}, \cos ^{-1} \frac{6}{7}\) and \(\cos ^{-1} \frac{2}{7}\)

**Question 14. If \(\vec{a}+\vec{b}=\vec{c}\), then the angle between \(\vec{a}\) and \(\vec{b}\) is**

- 90°
- 180°
- 120°
- Zero

**Answer:** 4. Zero

**Question 15. If at and are two non-collinear unit vectors and if \(\left|a_1+a_2\right|=\sqrt{3}\) then the value of \(\left(a_1-a_2\right) \cdot\left(2 a_1+a_2\right)\) is**

- 2
- 3/2
- 1/2
- 1

**Answer:** 3. 1/2

**Question 16. Which of the following is a vector quantity?**

- Temperature
- Impulse
- Gravitational potential
- Power

**Answer:** 2. Impulse

**Question 17. The resultant of two forces of magnitude (x+y) and (x-y) is \(\sqrt{x^2+y^2}\). The angle between them is**

- \(\cos ^{-1}\left[-\frac{\left(x^2+y^2\right)}{2\left(x^2-y^2\right)}\right]\)
- \(\cos ^{-1}\left[-\frac{2\left(x^2-y^2\right)}{x^2+y^2}\right]\)
- \(\cos ^{-1}\left[-\frac{x^2+y^2}{x^2-y^2}\right]\)
- \(\cos ^{-1}\left[-\frac{x^2-y^2}{x^2+y^2}\right]\)

**Answer:** 1. \(\cos ^{-1}\left[-\frac{\left(x^2+y^2\right)}{2\left(x^2-y^2\right)}\right]\)

**Question 18. ABC is an equilateral triangle and O is its centre of mass. Find n if \(\overrightarrow{A B}+\overrightarrow{A C}=n \overrightarrow{A O}\)**

- 1
- 2
- 3
- 4

**Hint:** \(\overrightarrow{A B}+\overrightarrow{A C}=(\overrightarrow{A O}+\overrightarrow{O B})+(\overrightarrow{A O}+\overrightarrow{O C})\)

= \(2 \overrightarrow{A O}+(\overrightarrow{O B}+\overrightarrow{O C})=2 \overrightarrow{A O}+\overrightarrow{A O}\)

= \(3 \overrightarrow{A O}\)

**Answer:** 3. 3

**Question 19. The sum of the magnitudes of two forces acting at a point is 16 N. The resultant has a magnitude of 8N and is perpendicular to the force of lower magnitude. The two forces are**

- 6N and 10N
- 8N and 8N
- 4N and 12N
- 2N and 14N

**Answer:** 1. 6N and 10N

**Question 20. Two men have velocities of 4 m/s towards the east and 3 m/s towards west, respectively. The velocity of the first man relative to the second is**

- \((4 \hat{i}+3 \hat{j})\)
- \((3 \hat{i}+4 \hat{j})\)
- \((4 \hat{i}-3 \hat{j})\)
- \((3 \hat{i}-4 \hat{j})\)

**Answer:** 1. \((4 \hat{i}+3 \hat{j})\)

**Question 21. Two forces in the ratio 1:2 act simultaneously on a particle. The result of these forces is three times the first force. The angle between them is**

- 0°
- 60°
- 90°
- 45°

**Answer:** 1. 0°

**Question 22. Given \(\vec{A}=2 \hat{i}+3 \hat{j} \text { and } \vec{B}=\hat{i}+\hat{j}\). The component of vector \(\vec{A}\) along vector \(\vec{B}\) is**

- \(\frac{1}{\sqrt{2}}\)
- \(\frac{3}{\sqrt{2}}\)
- \(\frac{5}{\sqrt{2}}\)
- \(\frac{7}{\sqrt{2}}\)

**Answer:** 3. \(\frac{5}{\sqrt{2}}\)

**Question 23. One of the components of a velocity vector of magnitude 50 m · s ^{-1} Is 30 m · s^{-1}, Its other orthogonal component Is**

- 15 m · s
^{-1} - 20 m · s
^{-1} - 25 m · s
^{-1} - 40 m · s
^{-1}

**Answer**: 4. 40 m · s^{-1}

**Question 24. The times of flight of the two projectiles are t _{1} and t_{2} If R is the horizontal range of each of them, then**

- \(t_1 t_2 \propto R\)
- \(t_1 t_2 \propto R^2\)
- \(t_1 t_2 \propto R^3\)
- \(t_1 t_2 \propto R^{\frac{1}{2}}\)

**Hint:** The angles of projection are \(\theta\) and \(\left(90^{\circ}-\theta\right)\).

∴ \(t_1 t_2=\frac{2 u \sin \theta}{g} \cdot \frac{2 u \sin \left(90^{\circ}-\theta\right)}{g}\)

= \(\frac{4 u^2}{g^2} \sin \theta \cos \theta=\frac{2}{g} \cdot \frac{u^2 \sin 2 \theta}{g}=\frac{2}{g} R\)

∴ \(t_1 t_2\propto R\)

**Answer:** 1. \(t_1 t_2 \propto R\)

**Question 25. For angles θ and (90° – θ) of projection, a projectile has the same horizontal range R. The maximum heights attained are H _{1} and H_{2} respectively. Then the relation among R, H_{1,} and H_{2} is**

- \(R=\sqrt{H_1 H_2}\)
- \(R=\sqrt{H_1^2+H_2^2}\)
- \(R=H_1+H_2\)
- \(R=4 \sqrt{H_1 H_2}\)

**Answer**: 4. \(R=4 \sqrt{H_1 H_2}\)

**Question 26. An airplane is flying with a velocity of 216 km/h at an altitude of 1960 m relative to the ground. It drops a bomb when it is just above point A on the ground. The bomb hits the ground at B. The distance AB is**

- 1.2 km
- 0.33 km
- 3.33 km
- 33 km

**Answer**: 1. 1.2 km

**Question 27. The initial velocity and the acceleration of a particle are \(\vec{u}\) = 3\(\hat{i}\)+4\(\hat{j}\) and \(\vec{a}\) = 0.3 \(\hat{i}\)+0.4\(\hat{j}\). The magnitude of its velocity after 10 s is**

- 10 units
- 8.5 units
- 77√2 units
- 7 units

**Answer:** 1. 10 units

**Question 28. The equations of motion of a projectile are x = 36t and 2y = 96t- 9.8 t². The angle of projection is**

- \(\sin ^{-1}\left(\frac{4}{5}\right)\)
- \(\sin ^{-1}\left(\frac{3}{5}\right)\)
- \(\sin ^{-1}\left(\frac{3}{4}\right)\)
- \(\sin ^{-1}\left(\frac{4}{3}\right)\)

**Hint:** x= 36t and y = 48t – \(\frac{1}{2}\) x 9.8t²—they suggest that the horizontal and vertical components of initial velocity are 36 and 48 units, respectively.

∴ \(\tan \theta=\frac{48}{36}=\frac{4}{3} \text { or, } \sin \theta=\frac{4}{5}\)

**Answer:** 1. \(\sin ^{-1}\left(\frac{4}{5}\right)\)

**Question 29. Two projectiles, projected with angles (45°-θ) and (45° +θ) respectively, have their horizontal ranges in the ratio**

- 2:1
- 1:1
- 2:3
- 1:2

**Answer:** 2. 1:1

**Question 30. For a projectile, (horizontal range)² = 48 x (maximum height)². The angle of projection is**

- 45°
- 60°
- 75°
- 30°

**Answer:** 4. 30°

**Question 31. Two railway tracks are parallel to the west-east direction. Along one track train A moves with a speed of 30 m · s ^{-1} from west to east, while along the second track, train B moves with a speed of 48 m · s^{-1} from east to west. The relative speed of B with respect to A is**

- 48 m · s
^{-1} - -78 m · s
^{-1} - 30 m · s
^{-1} - Zero

**Answer:** 2. 48 m · s^{-1}

**Question 32. Angle between the vectors \(\hat{i}+\hat{j} \text { and } \hat{i}-\hat{k}\) is**

- 60°
- 30°
- 45°
- 90°

**Answer:** 1. 60°

**Question 33. A vector is multiplied by (-2). As a result**

- The magnitude of the vector is doubled and the direction is unaltered
- The magnitude of the vector remains the same and the direction is reversed
- The magnitude of the vector is doubled and the direction is reversed
- No change in the magnitude or direction of the vector

**Answer:** 3. The Magnitude of the vector is doubled and the direction is reversed

**Question 34. In a clockwise system**

- \(\hat{j} \times \hat{j}=1\)
- \(\hat{k} \cdot \hat{i}=1\)
- \(\hat{j} \times \hat{k}=\hat{i}\)
- \(\hat{i} \cdot \hat{i}=0\)

**Answer:** 3. \(\hat{j} \times \hat{k}=\hat{i}\)

**Question 35. A vector \(\vec{P}=3 \hat{i}-2 \hat{j}+a \hat{k}\) is perpendicular to the vector \(\vec{Q}=2 \hat{i}+\hat{j}-\hat{k}\). The value of a is**

- 2
- 1
- 4
- 3

**Answer:** 3. 4

**Question 36. For any two vectors \(\vec{A} \text { and } \vec{B} \text {, if } \vec{A} \cdot \vec{B}=|\vec{A} \times \vec{B}|\), the magnitude of \(\vec{C}=\vec{A}+\vec{B}\) is equal to**

- \(\sqrt{A^2+B^2}\)
- \(A+B\)
- \(\sqrt{A^2+B^2+\frac{A B}{\sqrt{2}}}\)
- \(\sqrt{A^2+B^2+\sqrt{2} A B}\)

**Answer:** 4. \(\sqrt{A^2+B^2+\sqrt{2} A B}\)

**Question 38. A vector perpendicular to both of \((3 \hat{i}+\hat{j}+2 \hat{k})\) and \((2 \hat{i}- 2 \hat{j}+4 \hat{k})\) is**

- \(\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})\)
- \(\hat{i}-\hat{j}-\hat{k}\)
- \(\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})\)
- \((\sqrt{3} \hat{i}-\hat{j}-\hat{k})\)

**Answer:** 2. \(\hat{i}-\hat{j}-\hat{k}\)

**Question 39. A vector normal to \(a \cos \theta \hat{i}+b \sin \theta \hat{j}\) is**

- \(b \sin \theta \hat{i}-a \cos \theta \hat{j}\)
- \(\frac{1}{a} \sin \theta \hat{i}-\frac{1}{b} \cos \theta \hat{j}\)
- \(5 \hat{k}\)
- All of the above

**Answer:** 4. All of the above

**Question 40. A vector \(\vec{A}\) of magnitude 2 units is inclined at angles 30° and 60° with positive x- and y-axes, respectively. Another vector of magnitude 5 units is aligned along the positive x-axis. Then \(\vec{A}\) • \(\vec{B}\) is**

- 5√3
- 3√5
- 2√3
- 3√2

**Answer:** 1. 5√3

**Question 41. For what values of a and b, the vector \((a \hat{i}+b \hat{j})\) will be a unit vector perpendicular to the vector \((\hat{i}+\hat{j})\)?**

- 1,0
- 0,1
- \(\frac{1}{\sqrt{3}},-\frac{2}{\sqrt{3}}\)
- \(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\)

**Answer:** 4. \(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\)

**Question 42. Two billiard balls, starting from the same point, have velocities \((\hat{i}+\sqrt{3} \hat{j}) \text { and }(2 \hat{i}+2 \hat{j})\), respectively. The angle between them is**

- 60°
- 15°
- 45°
- 30°

**Answer:** 2. 15°

**Question 43. If \(|\vec{A} \times \vec{B}|=\sqrt{3} \vec{A} \cdot \vec{B}\), then \(|\vec{A}+\vec{B}|=\)?**

- \(\left(A^2+B^2+A B\right)^{\frac{1}{2}}\)
- (\(\left(A^2+B^2+\frac{A B}{\sqrt{3}}\right)^{\frac{1}{2}}\)
- A+B
- \(\left(A^2+B^2+\sqrt{3} A B\right)^{\frac{1}{2}}\)

**Answer**: 1. \(\left(A^2+B^2+A B\right)^{\frac{1}{2}}\)

**Question 44. The vector product of \(\vec{A}\) and \(\vec{B}\) is zero. The scalar product of \(\vec{A}\) and \((\vec{A}+\vec{B})\) is**

- 0
- \(A^2\)
- \(A B\)
- \(A^2+A B\)

**Answer:** 4. \(A^2+A B\)

**Question 45. If \(\vec{A} \cdot \vec{B}=\vec{A} \cdot \vec{C}=0\), then the vector parallel to \(\vec{A}\) would be**

- \(\vec{C}\)
- \(\vec{B}\)
- \(\vec{B} \times \vec{C}\)
- \(\vec{A} \times(\vec{B} \times \vec{C})\)

**Answer:** 3. \(\vec{B} \times \vec{C}\)

**In this type of question, more than one option is correct.**

**Question 46. In the following figure which of the statements is correct?**

- The sign of x -component of \(\vec{l}_1\) and \(\vec{l}_2\) is negative
- The signs of the y -component of \(\vec{l}_1 \text { and } \vec{l}_2\) are positive and negative respectively
- The signs of x and y-components of \(\vec{l}_1+\vec{l}_2\) are both positive
- None of these

**Answer:**

1. The sign of x -component of \(\vec{l}_1\) and \(\vec{l}_2\) is negative

3. The signs of x and y-components of \(\vec{l}_1+\vec{l}_2\) are both positive

**Question 47. Two particles are projected in the air with speed v _{0} at angles θ_{1} and θ_{2} (both acute) to the horizontal, respectively. If the height reached by the first particle is greater than that of the second, then which are the correct choices?**

- The angle of projection: θ
_{1 }> θ_{2} - Time of flight: T
_{1}> T_{2} - Horizontal range: R
_{1}> R_{2} - Total energy: U
_{1}> U_{2}

**Answer:**

**Question 48. For two vectors \(\vec{A} \text { and } \vec{B},|\vec{A}+\vec{B}|=|\vec{A}-\vec{B}|\) is always true when**

- \(|\vec{A}|=|\vec{B}| \neq 0\)
- \(\vec{A} \perp \vec{B}\)
- \(|\vec{A}|=|\vec{B}| \neq 0\) and \(\vec{A}\) and \(\vec{B}\) are parallel or antiparallel
- When either \(|\vec{A}|\) or \(|\vec{B}|\) is zero

**Answer:**

2. \(\vec{A} \perp \vec{B}\)

4. When either \(|\vec{A}|\) or \(|\vec{B}|\) is zero

**Question 49. Given \(\vec{a}+\vec{b}+\vec{c}+\vec{d}=\overrightarrow{0}\). Which of the following statements is correct?**

- \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) must each be a null vector
- The magnitude of \((\vec{a}+\vec{c})\) equals the magnitude of \((\vec{b}+\vec{d})\)
- The magnitude of \(\vec{a}\) can never be greater than the sum of the magnitudes of \(\vec{b}, \vec{c}\), and \(\vec{d}\)
- \((\vec{b}+\vec{c})\) must lie on the plane of \(\vec{a}\) and \(\vec{d}\) if \(\vec{a}\) and \(\vec{d}\) are not collinear and in the line of \(\vec{a}\) and \(\vec{d}\), if they are collinear

**Answer:**

2. The magnitude of \((\vec{a}+\vec{c})\) equals the magnitude of \((\vec{b}+\vec{d})\)

3. The magnitude of \(\vec{a}\) can never be greater than the sum of the magnitudes of \(\vec{b}, \vec{c}\) and \(\vec{d}\)

4. \((\vec{b}+\vec{c})\) must lie on the plane of \(\vec{a}\) and \(\vec{d}\) if \(\vec{a}\) and \(\vec{d}\) are not collinear and in the line of \(\vec{a}\) and \(\vec{d}\), if they are collinear

**Question 50. State which of the following statements are false. A scalar quantity is one that**

- Is conserved in a process
- Can never take negative values
- Must be dimensionless
- Has the same value for observers with different orientations of axes

**Answer:**

- Is conserved in a process
- Can never take negative values
- Must be dimensionless

**Question 51. The resultant of \(\vec{P}\) and \(\vec{Q}\) is \(\vec{R}\). The angle between \(\vec{P}\) and \(\vec{Q}\) is**

- 90°, if R² = P² + Q²
- Less than 90°, if R2 > (P² + Q²)
- Greater than 90°, if R² < (P2 + Q²)
- Greater than 90°, if R² > (P² + Q²)

**Answer:**

- 90°, if R² = P² + Q²
- Less than 90°, if R2 > (P² + Q²)
- Greater than 90°, if R² < (P2 + Q²)

**Question 52. The magnitude of the vector product of A and B may be**

- Greater than AB
- Equal to AB
- Less than AB
- Zero

**Answer:**

2. Equal to AB

3. Less than AB

4. Zero

**Question 53. If \(\vec{A}=5 \hat{i}+6 \hat{j}+3 \hat{k} \text { and } \vec{B}=6 \hat{i}-2 \hat{j}-6 \hat{k}\), then**

- \(\vec{A}\) and \(\vec{B}\) are perpendicular
- \(\vec{A} \times \vec{B}=\vec{B} \times \vec{A}\)
- \(\vec{A}\) and \(\vec{B}\) have the same magnitude
- \(\vec{A} \cdot \vec{B}=0\)

**Answer:**

1. \(\vec{A}\) and \(\vec{B}\) are perpendicular

4. \(\vec{A} \cdot \vec{B}=0\)

**Question 54. Two projectiles, thrown from the same point with the same magnitude of velocity, have angles 60° and 30° of their projection. Then their**

- Maximum heights attained are the same
- Horizontal ranges are the same
- Magnitudes of velocity at the instants of hitting the ground are the same
- Times of flight are the same

**Answer:**

2. Horizontal ranges are the same

3. Magnitudes of velocity at the instants of hitting the ground are the same

**Question 55. A body is projected with a velocity u at an angle θ with the horizontal. At t = 2s, the body makes an angle of 30° with the horizontal. 1 s later, it attains its maximum height. Then**

- u = 20√3 m/s
- θ = 60°
- θ = 45
- u = \(\frac{20}{\sqrt{3}} \mathrm{~m} / \mathrm{s}\)

**Answer:**

- u = 20√3 m/s
- θ = 60°

**Question 56. \(\vec{A} \perp \vec{B}\); \(\vec{C}\) is coplanar with \(\vec{A}\) and \(\vec{B}\). Therefore,**

- \(\vec{A}=x \vec{B}+y \vec{C}\), where x and y are scalars
- \(\vec{A} \cdot(\vec{B} \times \vec{C})=0\)
- \(|(\vec{A} \times \vec{B}) \times \vec{C}|=A B C\)
- \(\vec{A} \cdot \vec{B}=0\)

**Answer:**

- \(\vec{A}=x \vec{B}+y \vec{C}\), where x and y are scalars
- \(\vec{A} \cdot(\vec{B} \times \vec{C})=0\)
- \(|(\vec{A} \times \vec{B}) \times \vec{C}|=A B C\)
- \(\vec{A} \cdot \vec{B}=0\)