Short Questions:
Q.1. Find the magnitude (modulus) of given vector $\vec{a} = 3\hat{i} – 2\hat{j} + \hat{k}$.
Ans: $\sqrt{14}$ unit.
Q.2. Find the unit vector along (of) $\vec{a} = 2\hat{i} – \hat{j} +2\hat{k}$.
Ans: 3 unit.
Q.3. Find the magnitude of $\vec{b} = \frac{2}{3}\hat{i} – \frac{1}{3}\hat{j} +\frac{2}{3}\hat{k}$.
Ans: 1
Q.4. If $\vec{A} = 4\hat{i} – 3\hat{j}$ and $\vec{B} = 7\hat{i} +5\hat{j} + \hat{k}$, find the angle between the vectors $\vec{A}$ and $\vec{B}$.
Ans: 54°
Q.5. $\vec{A}$ and $\vec{B}$ are two non-zero vectors. If $|\vec{A} \times \vec{B}| = |\vec{A}. \vec{B}|$, what is the angle between $\vec{A}$ and $\vec{B}$?
Ans: 45°
OR
Q.6. If the scalar product of two vectors in equal to the magnitude of their vector product, find the angle between them.
Ans: 45°
Q.7. If $\vec{A}. \vec{B} = 0$, what is the value of angle between $\vec{A}$ and $\vec{B}$ ?
Ans: 90°
Q.8. Resultant of two equal forces may have the magnitude equal to one of the forces.
At what angle between the two equal forces this is possible. Justify your answer.
Ans: 120°
OR
Q. Two vectors have equal magnitudes and their resultant has the same magnitude.
What is the angle between the two vectors?
Ans: 120°
OR
Q. Can the sum of two equal vectors be equal to either vector?
Ans: Yes at $\theta = 120°$
OR
Q. What should be the angle between the two vectors of same magnitude,
so that their resultant is equal to either of them?
Ans: 120°
Q.9. Two vectors are given as $\vec{V}_1 = 2\hat{i} + 3\hat{j} + 4\hat{k}$ and $\vec{V}_2 = 3\hat{i} + 2\hat{j} – 4\hat{k}$.
Which one of two is larger in magnitude? Justify your answer.
Ans: Equal Magnitude = $\sqrt{29}$ units
Q.10. A vector is defined as $\vec{E} = 2\hat{i} + 3\hat{j} – 4\hat{k}$. What is the magnitude of the Y-component of $\vec{E}$?
Ans: 3
Q.11. Given two vectors $\vec{A} = 4.00\hat{i} +3.00\hat{j}$ and $\vec{B} = 5.00\hat{i} +2.00\hat{j}$. Find the magnitude of each vector.
Ans: 5, $\sqrt{29}$
Q.12. What is the difference between scalar and vector products of two vectors? Explain.
Q.13. Is a physical quantity having magnitude and direction necessarily a vector quantity? Explain.
Q.14. A force (in Newton) expressed in vector notation as $\vec{F} = 2\hat{i} + \hat{j} – 3\hat{k}$ is applied on a body so that the
displacement produced in meter is given by $\vec{D} = \hat{i} – 2\hat{j} – 3\hat{k}$. Express the result and nature of the work done.
Ans: 9 J
Q.15. A force (in Newton) expressed in vector notation as $\vec{F} = 4\hat{i} + 7\hat{j} – 3\hat{k}$ is applied on a body and produces a
displacement (in meter), $\vec{D} = 3\hat{i} – 2\hat{j} – 5\hat{k}$ in 4 seconds. Estimate the power.
Ans: 3.25 W
Q.16. If $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along x, y and z-axis respectively, find $\hat{i} \times (\hat{j} \times \hat{k})$.
Ans: 1
Q.17. If a vector has zero magnitude, is it meaning full to call it vector?
Q.18. The magnitude of two vectors are 3 and 4, and their dot product is 6, what is the angle between them?
Ans: 60°
Q.19. Two vectors $\vec{A}$ and $\vec{B}$ are such that $|\vec{A} – \vec{B}| = C$ and $|\vec{A} – \vec{B}| = C$. Find the angle between them.
Ans: 0°
Q.20. If a vector has zero magnitude, is it meaning full to call it vector?
Additional Short Questions:
Q.21. A vector $\vec{F} = \hat{i} + 2\hat{j} – 3\hat{k}$ is given. What is the magnitude of the y-component of the vector?
Ans: 2
Q.22. $\vec{C}$ is the vector sum of $\vec{A}$ and $\vec{B}$ i.e. $\vec{C} = \vec{A} + \vec{B}$ for $\vec{C}= \vec{A} + \vec{B}$ to be true, What is the angle between $\vec{A}$ and $\vec{B}$?
Ans: 0°
Q.23. If $\vec{A} = 4\hat{i} – 3\hat{j}+\hat{k}$ and $\vec{B} = 7\hat{i} +5\hat{j} + \hat{k}$, find the angle between the vectors $\vec{A}$ and $\vec{B}$.
Ans: 72.53°
Q.24. Can you find a vector quantity that has a magnitude of zero but components that are different from zero? Explain.
Q.25. What does $\vec{A}. \vec{A}$, the scalar product of a vector with itself gives?, What about $\vec{A} \times \vec{A}$, the vector product of a vector with itself?
Q.26. The angle between two vectors $\vec{A}$ and $\vec{B}$ is $\theta$. Find the magnitude and direction of $\vec{A} \times \vec{B}$ and $\vec{A}. \vec{B}$.
Q.27. If $\vec{A}$ and $\vec{B}$ are non zero vectors, is it possible for $\vec{A} \times \vec{B}$ and $\vec{A}. \vec{B}$ both to be zero? Explain.
Q.28. If $\vec{B}$ is added to $\vec{A}$, under what condition does the resultant vector have a magnitude equal to $A + B$? Under what conditions is the resultant vector equal to zero?
Ans: zero degree, if $A = B$ and $\theta= 180°$
Q.29. What is the angle between a two Newton force and a three Newton force so that their sum is four Newton?
Ans: 75.62°
Q.30. Two equal velocities have a resultant equal to 3/2 times the value of either velocity, find the angle between?
Ans: 82°-49′
Q.31: A force is acting at an angle of 30°to the X-axis. If the magnitude of force along Y-axis is 10 N, find the magnitude of force along X – axis.
Ans: $10\sqrt{3}$
Q.32. The force is acting towards xy – plane at an angle of 30º with Y – axis. If the magnitude of force along X-axis is 20N, find the magnitude of force along y axis.
Ans: $20\sqrt{3}$ N
Q.33 Two equal velocities have a resultant equal to 3/2 times the value of either velocity, find the angle between?
Ans: 82°-49′
Q.34. What is the angle between a two Newton force and a three Newton force so that their sum is four Newton ?
Ans: 75.52°
Q.35. Two vectors having equal magnitudes A makes an angle $\theta$ with each other. Find the magnitude and direction of the resultant.
Ans: $2A\cos\frac{\theta}{2}$ and $\frac{\theta}{2}$
Q.36. The angle between two vectors of equal magnitude is 120°. Prove that the magnitude of the resultant is equal to either of them.
Q.37. Is it possible that $\vec{A} -\vec{B}=\vec{A}+\vec{B}$?
Ans: Yes when $\vec{B} = 0$
Q.38. What is the angle between $(\hat{i} + \hat{j})$ and $(\hat{i} – \hat{j})$?
Ans: 90°
Q.39. The resultant of two equal forces acting at right angle to each other is 1414 N. Find the magnitude of each force?
Ans: 1000 N
Numerical Questions
Q.1. A disoriented physics professor drives 3.25 km north, then 4.75 km west and then 1.50 km south. Find the magnitude and direction of the resultant displacement.
Ans: 5.06km, 20.22° north of west (or 69.78° W of N).
Q.2. A rocket fires two engines simultaneously. One produces a thrust of 725N directly forward while the other gives a 513N thrust at 32.4° above the forward direction. Find the magnitude and direction of the resultant force that these engines exerts on the rocket.
Ans: 1190.31N & 13.35°
Q.3. If the resultant of two forces (P + Q) and (P – Q) is $\sqrt{3P^2 + Q^2}$, find the angle between them.
Ans. $\theta = 60°$.
Additional Numerical Questions:
Q.4 Two forces $\vec{F}_1$ and $\vec{F}_2$ act upon a body in such a manner that the resultant force $\vec{R}$ has the magnitude equal to that of $\vec{F}_1$ and makes angle of 90º with $\vec{F}_1$. Let $F_1 = R = 10$ N. Find the magnitude of second force and its direction.
Ans: 14.14 N, 135°
Q.5 Find the projection of the vector $\vec{A} = 2\hat{i} +3\hat{j} + 2\hat{k}$ on the vector $\vec{B} = \hat{i} +2\hat{j} + \hat{k}$.
Ans: $10/\sqrt{6}$ unit
Q.6. A cave explorer is surveying a cave. He follows a passage 100m straight east, then 50m in a direction 30° west of north, then 150m at 45º west of south. After a fourth unmeasured displacement, he finds himself back where he started. Using a scale drawing to determine the forth displacement (magnitude and direction).
Ans: 70.02m in the direction 26.34° east of north.
Q.7 A spelunker is surveying a cave. She follows a passage of 180m straight west, then 210m in a direction 45° east of south, and then 280m at 30° east of north. After a fourth unmeasured displacement she finds herself back where she started. Use the method of components to determine the magnitude and direction of the fourth displacement.
Ans: 143.54 m 40.9° S of W.
Long Questions (Derivation):
Q.1. State the parallelogram law of vector addition. Derive expressions for the magnitude and direction of the resultant of two vectors inclined at an angle $\theta$ from each other.
Q.2 State triangle law of vector addition. Obtain an expression for the resultant of two vectors P and Q inclined at angle $\theta$.
Multiple Choice Questions:
- Which of the following statement about scalar is true ?
(a) It must have magnitude.
(b) Some scalar have both magnitude and direction.
(c) They don’t obey the law of vector addition
(d) All of above - Which of the following quantity is not a vector?
(a) Work
(b) Force
(c) acceleration
(d) displacement - Which of the following quantity has both magnitude and direction but not vector?
(a) Velocity
(b) Force
(c) acceleration
(d) time - Electric current and time have magnitude and direction but not vectors because
(a) They obey law of vector addition.
(b) They don’t obey the law of vector addition.
(c) They obey the law of vector multiplication.
(d) They obey the law of vector division. - Which of the following isn’t a scalar?
(a) Energy
(b) speed
(c) Pressure
(d) field strength - Which of the following is a vector?
(a) Torque
(b) mass
(c) distance
(d) volume - Which of the following is not a vector?
(a) Velocity
(b) charge
(c) Field intensity
(d) temperature gradient - Which is a scalar quantity?
(a) Gravitational field intensity
(b) Pressure
(c) Retardation
(d) Upthrust - The possible resultant of two forces 20 N and 10 N is
(a) 8 N
(b) 27 N
(c) 40 N
(d) 200 N
Ans: b - Which is not a vector?
(a) Displacement
(b) Force
(c) Acceleration
(d) Energy
Additional Multiple Choice Questions:
- A vector remains unchanged
(a) when it is rotated by an arbitrary angle.
(b) when it is cross multiplied by a unit vector
(c) when it is multiplied by a scalar
(d) when it is shifted parallel to itself - Two vectors $\vec{A}$ and $\vec{B}$ are such that $|\vec{A} – \vec{B}| = C$ and $A – B = C$. Find the angle between them.
(a) 0°
(b) 45°
(c) 90°
(d) 180° - The resultant of two forces 20 N and 10 N can not be
(a) 10 N
(b) 20 N
(c) 30 N
(d) 35 N - If $A = B$ then
(a) $A = B$
(b) $\vec{A} = \vec{B}$
(c) both a and b
(d) neither a nor b - The condition for vector $\vec{A} + \vec{B} = \vec{A} – \vec{B}$ is that:
(a) $\vec{A} = 0$
(b) $\vec{B} = 0$
(c) $\vec{A} = \vec{B}$
(d) $\vec{B}$ is a unit vector - What will be the maximum magnitude of $\vec{A} – \vec{B}$ ?
(a) $A + B$
(b) $A-B$
(c) $\sqrt{A^2 + B^2}$
(d) $\sqrt{A^2-B^2}$ - A vector is defined as $\vec{E} = 2\hat{i} + 3\hat{j} – 4\hat{k}$. What is the magnitude of the Y-component of $\vec{E}$?
(a) 2
(b) 3
(c) 4
(d) $\sqrt{29}$ - The dot product of two vectors is $\sqrt{3}$ times the magnitude of their cross product. Then angle between these two vectors will be
(a) 30°
(b) 45°
(c) 60°
(d) 75° - If angle between two equal forces (F) is 90° magnitude of the resultant is
(a) $\sqrt{2}$ F
(b) 2F
(c) F
(d) $2\sqrt{2}$F
Answer Key:
| 1. d | 2. a | 3. d | 4. b | 5. d | 6. a | 7. d | 8. b | 9. b | 10. d |
|---|---|---|---|---|---|---|---|---|---|
| 11. d | 12. a | 13. d | 14. c | 15. b | 16. a | 17. b | 18. a | 19. a |