The angle between two vectors plays an important role when the two vectors are multiplied. Following are the two ways through a vector multiplication can be carried out:
- Multiplication of a Vector by a scalar
- Multiplication of a vector by a vector
The multiplication of a vector by a vector can be carried out in two ways and they are termed as:
- Dot Product
- Cross Product
The result of a dot product of two vectors is a scalar quantity. The result of the cross product of two vectors is a vector quantity.
Question and Answer
1. Two vectors \(2\hat{i}+3\hat{j}+8\hat{k}\) and \(4\hat{j}-4\hat{j}+\alpha\hat{k}\) are perpendicular to each other. What is the value of \(\alpha\)?
- ½
- -½
- 1
- -1
Answer: (b) -½
2. A force of 50 N acts on a body and displaces it through a distance of 10 meters in a direction making an angle of 600 with the force. What is the work done by the force?
- 100 J
- 150 J
- 200 J
- 250 J
Answer: (d) 250 J
3. If the sum of two vectors is perpendicular to the difference between them. The ratio of their magnitude will be
- 1
- 2
- 3
- 4
Answer: (a) 1
4. If \(\vec{A}\times \vec{B}=\vec{A}.\vec{B}\) then the angle between A and B will be
- π
- π/2
- π/3
- π/4
Answer: (d) π/4
5. What is the value of \(\left | \vec{P} \times \vec{Q}\right |\) if \(\vec{P}=3\hat{i}+\hat{j}+2\hat{k}\) and \(\vec{Q}=2\hat{i}-2\hat{j}+4\hat{k}\)?
- \(8\sqrt{3}\)
- \(3\sqrt{8}\)
- \(4\sqrt{3}\)
- \(3\sqrt{4}\)
Answer: (a) \(8\sqrt{3}\)
6. If \(\vec{P}\times\vec{Q}=\vec{R}\), then which of the following statements is true?
- \(\vec{R}\perp \vec{P}\)
- \(\vec{R}\perp \vec{Q}\)
- \(\vec{R}\perp (\vec{P}+\vec{Q})\)
- \(\vec{R}\perp (\vec{P}\times \vec{Q})\)
Answer: (d) \(\vec{R}\perp (\vec{P}\times \vec{Q})\)
7. The magnitude of the scalar product of the vectors \(\vec{V_1}=2\hat{i}+5\hat{k}\) and \(\vec{V_2}=3\hat{j}+4\hat{k}\) is
- 20
- 36
- 24
- 34
Answer: (a) 20
8. Which of the following statements is true if \(\vec{P}\) and \(\vec{Q}\) are perpendicular to each other?
- \(\vec{P}+\vec{Q}=0\)
- \(\vec{P}-\vec{Q}=0\)
- \(\vec{P}\cdot \vec{Q}=0\)
- \(\vec{P}\times \vec{Q}=0\)
Answer: (c) \(\vec{P}\cdot \vec{Q}=0\)
9. What is the angle between the vectors \(-2\hat{i}+3\hat{j}+\hat{k}\) and \(\hat{i}+2\hat{j}+-4\hat{k}\)?
- \(0^{\circ}\)
- \(25^{\circ}\)
- \(90^{\circ}\)
- \(180^{\circ}\)
Answer: (c) \(90^{\circ}\)
10. A force \(\vec{F}=5\hat{i}+6\hat{i}+4\hat{k}\) acts on a body and displaces it by \(\vec{S}=6\hat{i}-5\hat{k}\). What is the work done by the force?
- 5 Units
- 10 Units
- 15 Units
- 20 Units
Answer: (b) 10 Units
11. What is the area of the parallelogram which represented by vectors \(\vec{P}=2\hat{i}+3\hat{j}\) and \(\vec{Q}={i}+4\hat{j}\)?
- 5 Units
- 10 Units
- 15 Units
- 20 Units
Answer: (a) 5 Units
12. What is the angle between vectors \(\vec{A}\times \vec{B}\) and \(\vec{B}\times \vec{A}\)?
- π
- 00
- π/2
- π/4
Answer: (a)
13. The angle between \(\vec{A}+\vec{B}\) and \(\vec{A}\times \vec{B}\) is
- π
- π/2
- π/4
- 0
Answer: (b) π/2
14. The vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) satisfy the relation \(\vec{a}\cdot \vec{b}=0\) and \(\vec{a}\cdot \vec{c}=0\). The vector \(\vec{a}\) is parallel to
- \(\vec{b}\cdot \vec{c}\)
- \(\vec{b}\times \vec{c}\)
- \(\vec{b}\)
- \(\vec{c}\)
Answer: (b) \(\vec{b}\times \vec{c}\)
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