vector Algebra, 20 mcqs on it with explained answers
Here are 20 multiple-choice questions (MCQs) on Vector Algebra with explained answers for each:
1. What is the dot product of vectors $$\mathbf{A} = 3\mathbf{i} + 2\mathbf{j}$$ and $$\mathbf{B} = \mathbf{i} - \mathbf{j}$$?
- a) 1
- b) 5
- c) 3
- d) -1
- Answer: a) 1. Explanation: Dot product = $$3 \times 1 + 2 \times (-1) = 3 - 2 = 1$$.
2. The cross product $$\mathbf{A} \times \mathbf{B}$$ results in a vector...
- a) parallel to $$\mathbf{A}$$
- b) parallel to $$\mathbf{B}$$
- c) perpendicular to both $$\mathbf{A}$$ and $$\mathbf{B}$$
- d) zero vector always
- Answer: c) perpendicular to both. Explanation: Cross product is perpendicular to the plane containing $$\mathbf{A}$$ and $$\mathbf{B}$$.
3. Magnitude of vector $$\mathbf{A} = 4\mathbf{i} - 3\mathbf{j}$$ is...
- a) 1
- b) 5
- c) 7
- d) 12
- Answer: b) 5. Explanation: $$\sqrt{4^2 + (-3)^2} = \sqrt{16+9} = \sqrt{25} = 5$$.
4. Two vectors are said to be orthogonal if their dot product is...
- a) zero
- b) one
- c) negative
- d) positive
- Answer: a) zero. Explanation: Orthogonal vectors have zero dot product.
5. If $$\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos\theta$$, then $$\theta$$ is the...
- a) angle between $$\mathbf{A}$$ and $$\mathbf{B}$$
- b) sum of angles of $$\mathbf{A}$$ and $$\mathbf{B}$$
- c) product of magnitudes
- d) difference of vectors
- Answer: a) angle between $$\mathbf{A}$$ and $$\mathbf{B}$$.
6. The vector triple product $$\mathbf{A} \times (\mathbf{B} \times \mathbf{C})$$ equals...
- a) $$(\mathbf{A} \times \mathbf{B}) \times \mathbf{C}$$
- b) $$\mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})$$
- c) zero vector
- d) $$\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$$
- Answer: b) $$\mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})$$. Explanation: This is the vector triple product identity.
7. The projection of vector $$\mathbf{A}$$ on vector $$\mathbf{B}$$ is given by...
- a) $$\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} \hat{\mathbf{B}}$$
- b) $$\mathbf{A} \times \mathbf{B}$$
- c) $$|\mathbf{A}| |\mathbf{B}|$$
- d) $$\mathbf{B} \times \mathbf{A}$$
- Answer: a) $$\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} \hat{\mathbf{B}}$$.
8. If two vectors are parallel, their cross product is...
- a) zero vector
- b) non-zero vector
- c) depends on angle
- d) undefined
- Answer: a) zero vector. Explanation: Cross product zero means vectors are parallel or one is zero.
9. Unit vector in the direction of $$\mathbf{A} = 6\mathbf{i} + 8\mathbf{j}$$ is...
- a) $$\frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j}$$
- b) $$6\mathbf{i} + 8\mathbf{j}$$
- c) $$\mathbf{i} + \mathbf{j}$$
- d) $$\frac{6}{8}\mathbf{i} + \frac{8}{6}\mathbf{j}$$
- Answer: a) $$\frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j}$$. Explanation: Magnitude = 10, unit vector divides components by 10.
10. Scalar triple product $$\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$$ gives...
- a) area of parallelogram
- b) volume of parallelepiped
- c) angle between vectors
- d) zero always
- Answer: b) volume of parallelepiped.
11. Vector $$\mathbf{A} = 2\mathbf{i} + 3\mathbf{j}$$ and $$\mathbf{B} = -4\mathbf{i} + \mathbf{j}$$, $$\mathbf{A} + \mathbf{B}$$ is...
- a) $$-2\mathbf{i} + 4\mathbf{j}$$
- b) $$-6\mathbf{i} + 2\mathbf{j}$$
- c) $$2\mathbf{i} + 4\mathbf{j}$$
- d) $$-6\mathbf{i} + 4\mathbf{j}$$
- Answer: a) $$-2\mathbf{i} + 4\mathbf{j}$$.
12. The position vector of the midpoint between points $$\mathbf{P}$$ and $$\mathbf{Q}$$ is...
- a) $$\mathbf{P} - \mathbf{Q}$$
- b) $$\frac{\mathbf{P} + \mathbf{Q}}{2}$$
- c) $$\mathbf{P} \times \mathbf{Q}$$
- d) $$\mathbf{P} + \mathbf{Q}$$
- Answer: b) $$\frac{\mathbf{P} + \mathbf{Q}}{2}$$.
13. Which of the following is a vector quantity?
- a) speed
- b) distance
- c) displacement
- d) time
- Answer: c) displacement.
14. The angle between two vectors $$\mathbf{A}$$ and $$\mathbf{B}$$ with $$\mathbf{A} \cdot \mathbf{B} = 0$$ is...
- a) 0°
- b) 45°
- c) 90°
- d) 180°
- Answer: c) 90°.
15. If $$\mathbf{A} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$$, the magnitude is...
- a) $$a + b + c$$
- b) $$a^2 + b^2 + c^2$$
- c) $$\sqrt{a^2 + b^2 + c^2}$$
- d) $$\sqrt{a + b + c}$$
- Answer: c) $$\sqrt{a^2 + b^2 + c^2}$$.
16. Which of these is true for any vectors $$\mathbf{A}$$, $$\mathbf{B}$$?
- a) $$\mathbf{A} \times \mathbf{B} = \mathbf{B} \times \mathbf{A}$$
- b) $$\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$$
- c) $$|\mathbf{A} + \mathbf{B}| = |\mathbf{A}| + |\mathbf{B}|$$
- d) $$\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$$ false
- Answer: b) $$\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$$.
17. The vector sum $$\mathbf{A} + \mathbf{B}$$ is commutative. It means...
- a) $$\mathbf{A} + \mathbf{B} = -(\mathbf{B} + \mathbf{A})$$
- b) $$\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$$
- c) $$\mathbf{A} + \mathbf{B} = 0$$
- d) None
- Answer: b) $$\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$$.
18. The vector $$\mathbf{A} - \mathbf{B}$$ is...
- a) vector from $$\mathbf{B}$$ to $$\mathbf{A}$$
- b) vector from $$\mathbf{A}$$ to $$\mathbf{B}$$
- c) scalar quantity
- d) zero vector
- Answer: a) vector from $$\mathbf{B}$$ to $$\mathbf{A}$$.
19. The angle between $$\mathbf{A}$$ and $$-\mathbf{A}$$ is...
- a) 0°
- b) 90°
- c) 180°
- d) depends on vectors
- Answer: c) 180°.
20. Vector addition follows which law?
- a) Associative law only
- b) Commutative law only
- c) Both associative and commutative laws
- d) Neither
- Answer: c) Both associative and commutative laws.
20 more
Here are 20 more multiple-choice questions on Vector Algebra with explained answers:
1. If $$\mathbf{A} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$$, then the unit vector along $$\mathbf{A}$$ is:
- a) $$\frac{1}{\sqrt{14}}(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$$
- b) $$\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$$
- c) $$\mathbf{i} + \mathbf{j} + \mathbf{k}$$
- d) $$\frac{1}{14}(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$$
- Answer: a) Unit vector is vector divided by its magnitude. Magnitude = $$\sqrt{1^2+2^2+3^2}=\sqrt{14}$$.
2. The scalar projection of $$\mathbf{A}$$ on $$\mathbf{B}$$ is given by:
- a) $$\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|}$$
- b) $$\mathbf{A} \times \mathbf{B}$$
- c) $$|\mathbf{A}||\mathbf{B}|\sin\theta$$
- d) $$|\mathbf{A}||\mathbf{B}|\cos\theta$$
- Answer: a) Scalar projection is the magnitude of the projection vector.
3. If $$\mathbf{A} \times \mathbf{B} = \mathbf{0}$$, then:
- a) $$\mathbf{A}$$ and $$\mathbf{B}$$ are perpendicular
- b) $$\mathbf{A}$$ and $$\mathbf{B}$$ are parallel
- c) $$\mathbf{A} = \mathbf{0}$$
- d) $$\mathbf{B} = \mathbf{0}$$
- Answer: b) Cross product zero means vectors are parallel or one is zero.
4. The vector $$\mathbf{A} \cdot \mathbf{B}$$ represents:
- a) vector product
- b) scalar product
- c) both
- d) neither
- Answer: b) Scalar product (dot product).
5. Volume of parallelepiped formed by $$\mathbf{A}, \mathbf{B}, \mathbf{C}$$ is:
- a) $$|\mathbf{A} \times \mathbf{B}|$$
- b) $$|\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})|$$
- c) $$|\mathbf{A} + \mathbf{B} + \mathbf{C}|$$
- d) $$|\mathbf{A} \cdot \mathbf{B} \cdot \mathbf{C}|$$
- Answer: b) Scalar triple product absolute value.
6. If $$\mathbf{A} = 3\mathbf{i} - \mathbf{j}$$ and $$\mathbf{B} = \mathbf{i} + 4\mathbf{j}$$, then $$\mathbf{A} \cdot \mathbf{B}$$ is:
- a) -1
- b) 7
- c) 3
- d) 0
- Answer: b) $$3 \times 1 + (-1) \times 4 = 3 -4 = -1$$, so answer corrected: a) -1.
7. The vector product $$\mathbf{A} \times \mathbf{B}$$ is anti-commutative. It means:
- a) $$\mathbf{A} \times \mathbf{B} = \mathbf{B} \times \mathbf{A}$$
- b) $$\mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A})$$
- c) $$\mathbf{A} \times \mathbf{B} = \mathbf{0}$$
- d) None
- Answer: b) Anti-commutative property.
8. The dot product of two perpendicular vectors is:
- a) 1
- b) 0
- c) -1
- d) depends on magnitude
- Answer: b) Zero.
9. The angle between vectors $$\mathbf{A}$$ and $$-\mathbf{A}$$ is:
- a) 0°
- b) 90°
- c) 180°
- d) none
- Answer: c) 180°.
10. The vector sum of $$\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}$$ and $$\mathbf{B} = -3\mathbf{i} + \mathbf{j}$$ is:
- a) $$6\mathbf{i} + 5\mathbf{j}$$
- b) $$0\mathbf{i} + 5\mathbf{j}$$
- c) $$0\mathbf{i} + 3\mathbf{j}$$
- d) $$6\mathbf{i} + 3\mathbf{j}$$
- Answer: b) Components add: $$3 - 3 = 0$$, $$4 + 1 = 5$$.
11. The magnitude of $$\mathbf{A} \times \mathbf{B}$$ equals:
- a) $$|\mathbf{A}||\mathbf{B}|\sin\theta$$
- b) $$|\mathbf{A}||\mathbf{B}|\cos\theta$$
- c) $$\mathbf{A} \cdot \mathbf{B}$$
- d) $$|\mathbf{A}| + |\mathbf{B}|$$
- Answer: a).
12. The direction of $$\mathbf{A} \times \mathbf{B}$$ is given by:
- a) right-hand rule
- b) left-hand rule
- c) parallel to $$\mathbf{A}$$
- d) parallel to $$\mathbf{B}$$
- Answer: a).
13. A vector with zero magnitude is called:
- a) zero vector
- b) unit vector
- c) null vector
- d) both a and c
- Answer: d).
14. If $$\mathbf{A} = 4\mathbf{i} - 4\mathbf{j}$$, then a vector perpendicular to $$\mathbf{A}$$ is:
- a) $$4\mathbf{j} + 4\mathbf{i}$$
- b) $$4\mathbf{j} + 4\mathbf{i}$$
- c) $$4\mathbf{j} + 4\mathbf{i}$$
- d) any vector satisfying $$\mathbf{A} \cdot \mathbf{X} = 0$$
- Answer: d).
15. Which vector has magnitude 1?
- a) $$\mathbf{A} = \mathbf{i} + \mathbf{j} + \mathbf{k}$$
- b) $$\mathbf{B} = \frac{\mathbf{i} + \mathbf{j} + \mathbf{k}}{\sqrt{3}}$$
- c) $$\mathbf{C} = 2\mathbf{i}$$
- d) $$\mathbf{D} = 0$$
- Answer: b).
16. The sum of two vectors is zero if:
- a) they are equal
- b) they are opposite and equal in magnitude
- c) one is zero vector
- d) magnitude sum is zero
- Answer: b).
17. The dot product of $$\mathbf{A} = 2\mathbf{i} + 3\mathbf{j}$$ and $$\mathbf{B} = -3\mathbf{i} + 4\mathbf{j}$$ is:
- a) 0
- b) 6
- c) 8
- d) -6
- Answer: d). Explanation: $$2 \times -3 + 3 \times 4 = -6 + 12 = 6$$, corrected to b) 6.
18. Which statement is true:
- a) $$\mathbf{A} \cdot \mathbf{B} = 0$$ implies $$\mathbf{A} \times \mathbf{B} = 0$$
- b) $$\mathbf{A} \times \mathbf{B} = 0$$ implies $$\mathbf{A} \cdot \mathbf{B} = 0$$
- c) $$\mathbf{A} \cdot \mathbf{B} = 0$$ implies $$\mathbf{A}$$ and $$\mathbf{B}$$ are perpendicular
- d) None
- Answer: c).
19. If $$\mathbf{A} = \mathbf{i} + \mathbf{j}$$, $$\mathbf{B} = \mathbf{j} + \mathbf{k}$$, then $$\mathbf{A} \times \mathbf{B}$$ is:
- a) $$\mathbf{i} + \mathbf{k}$$
- b) $$\mathbf{i} - \mathbf{k}$$
- c) $$\mathbf{i} \times \mathbf{k}$$
- d) $$\mathbf{0}$$
- Answer: b). Calculation: $$\mathbf{i}\times \mathbf{j} = \mathbf{k}$$, $$\mathbf{i}\times \mathbf{k} = -\mathbf{j}$$, etc.
20. The resultant of two vectors acting at a point is maximum when the angle between them is:
- a) 0°
- b) 90°
- c) 180°
- d) 45°
- Answer: a) 0°. Explained by vector addition magnitude maximization.
