1. Is a vector necessarily changed if it is rotated through an angle?
Answer: Yes. Rotating through an angle changes the direction of the vector.
Explanation: For two vectors to be equal their magnitudes and directions must be equal. In this case, magnitude remains the same but direction changes. So by rotating a vector, it changes.
2. Is it possible to add two vectors of unequal magnitudes and get zero? Is it possible to add three vectors of equal magnitudes and get zero?
Answer: Two vectors of unequal magnitudes cannot be added to get zero. Three vectors of equal magnitudes can be added to get zero.
Explanation: While adding two vectors we get minimum resultant value if their directions are opposite. The difference of the magnitudes is the magnitude of resultant. If two vectors have unequal magnitudes their difference cannot be zero.
Three vectors of equal magnitudes can be added to get zero. Think of these three vectors having their tails at a point and each having an angle of 120 degrees to others. In this situation, their resultant will be zero.
3. Does the phrase "direction of zero vector" have physical significance? Discuss in terms of velocity, force etc.
Answer: The phrase "direction of zero vector" does not have physical significance. Though the direction of the zero vector is indeterminate, even if you assign a direction to it, it will have no effect physically as it has zero magnitudes. If you give a body zero velocity or apply zero force in whatever direction it may be, it will not affect it physically.
4. Can you add three unit vectors to get a unit vector? Does your answer change if two unit vectors are along the co-ordinate axes?
Answer: Yes, three unit vectors can be added to get a unit vector. The answer remains the same even if two of them are along the co-ordinate axes.
Explanation: In fact, an unlimited number of combinations of three unit vectors can be had to get a sum of the unit vector. The only condition is that two of them have just opposite directions so they cancel out each other and the third unit vector is the resultant.
If two of the three unit vectors are along the co-ordinate axes then you can take the third unit vector opposite in direction to any of them and the resultant will be a unit vector.
5. Can we have physical quantities having magnitude and direction which are not vectors?
Answer: Yes, an electric current in a wire and flow of liquid in a pipe are some examples which have magnitude and direction but are not vectors.
Explanation: In fact only having magnitude and direction is not sufficient condition for being a vector but it is necessary that they follow the "Triangle rule of addition". Since electric current and flow of liquid in a pipe do not follow the "Triangle rule of addition" so they are not vectors.
6. Which of the following two statements is more appropriate?
(a) Two forces are added using triangle rule because the force is a vector quantity.
(b) Force is a vector quantity because two forces are added using triangle rule.
Answer: Statement (b) is more appropriate.
Explanation: explanation same as in question 5.
7. Can you add two vectors representing physical quantities having different dimensions? Can you multiply two vectors representing physical quantities having different dimensions?
Answer: Two vectors representing physical quantities having different dimensions cannot be added. But Two vectors representing physical quantities having different dimensions can be multiplied.
Explanation: Multiplication of vectors having different dimensions is common. Either as dot product or cross product.
8. Can a vector have zero component along a line and still have non-zero magnitude?
Answer: Yes. A non-zero magnitude vector has zero component along a line perpendicular to it.
Explanation: Component along a line is cosθ times the vector, where θ is the angle between the line and vector's direction. For θ=90° , Cosθ= 0. So the component along the line becomes zero.
9. Let ε₁ and ε₂ be the angles made by vectors A and -A with the positive X-axis. Show that tan ε₁=tan ε₂. Thus giving tan ε does not uniquely determine the direction of A.
Answer: Since A and - A are opposite in direction ie angle between them is 180°. so, in this case, ε₁= 180°+ ε₂. (See Diagram below)
∴ tan ε₁= tan(180°+ ε₂)= tan ε₂
Thus giving tan ε does not uniquely determine the direction of A because it indicates directions of both A and - A.
10. Is the vector sum of the unit vectors ⃗i and ⃗j a unit vector? If no, can you multiply this sum by a scalar number to get a unit vector?
Answer: Vector sum of the unit vectors ⃗i and ⃗j is not a unit vector because its magnitude is not unit. Yes, this sum can be multiplied by a scalar number to get a unit vector.
Explanation: Vectors ⃗i and ⃗j represent vectors of unit magnitudes along X-axis and Y-axis, so the angle between them is a right angle. Hence the magnitude of their vector sum will be √2. If we multiply this sum by a scalar number 1/√2 we can get a unit vector.
11. Let A=3i+4j. Write four vectors B such that A=/=B but A=B.
Answer: Four such vectors may be B=3i-4j, B=-3i+4j, B=-3i-4j and B= 5j
12. Can you have AxB=A.B with A=/=0 and B=/=0? What if one of the two vectors is zero?
Answer: If A=/=0 and B=/=0 then AxB cannot be equal to A.B because cross product is a vector quantity having a magnitude equal to ABsinθ and direction perpendicular to the plane containing A and B while the dot product A.B is a scalar quantity having its value equal to ABcosθ.
If one of the two vectors is zero then A.B=0 and magnitude of AxB is also zero. In this case, the direction of the zero vector has no significance.
13. If AxB=0, can you say that (a) A=B ,(b) A=/=B?
Answer: If AxB =0 then we can not say that (a) A=B or (b) A=/=B. Since AxB =ABsinθ, then if any of A, B or θ is zero then AxB =0 even if A=B or A=/=B.
14. If A=5i-4j and B= -7.5i+6j . Do we have B=kA? Can we say that B/ A=k?
Answer: In this case, we do have B=kA where k= -1.5. But we can not say that B/A=k because the division of vectors is not defined.
Answer: Yes, an electric current in a wire and flow of liquid in a pipe are some examples which have magnitude and direction but are not vectors.
Explanation: In fact only having magnitude and direction is not sufficient condition for being a vector but it is necessary that they follow the "Triangle rule of addition". Since electric current and flow of liquid in a pipe do not follow the "Triangle rule of addition" so they are not vectors.
6. Which of the following two statements is more appropriate?
(a) Two forces are added using triangle rule because the force is a vector quantity.
(b) Force is a vector quantity because two forces are added using triangle rule.
Answer: Statement (b) is more appropriate.
Explanation: explanation same as in question 5.
7. Can you add two vectors representing physical quantities having different dimensions? Can you multiply two vectors representing physical quantities having different dimensions?
Answer: Two vectors representing physical quantities having different dimensions cannot be added. But Two vectors representing physical quantities having different dimensions can be multiplied.
Explanation: Multiplication of vectors having different dimensions is common. Either as dot product or cross product.
8. Can a vector have zero component along a line and still have non-zero magnitude?
Answer: Yes. A non-zero magnitude vector has zero component along a line perpendicular to it.
Explanation: Component along a line is cosθ times the vector, where θ is the angle between the line and vector's direction. For θ=90° , Cosθ= 0. So the component along the line becomes zero.
9. Let ε₁ and ε₂ be the angles made by vectors A and -A with the positive X-axis. Show that tan ε₁=tan ε₂. Thus giving tan ε does not uniquely determine the direction of A.
Answer: Since A and - A are opposite in direction ie angle between them is 180°. so, in this case, ε₁= 180°+ ε₂. (See Diagram below)
 |
| Fig for Problem 9 |
∴ tan ε₁= tan(180°+ ε₂)= tan ε₂
Thus giving tan ε does not uniquely determine the direction of A because it indicates directions of both A and - A.
10. Is the vector sum of the unit vectors ⃗i and ⃗j a unit vector? If no, can you multiply this sum by a scalar number to get a unit vector?
Answer: Vector sum of the unit vectors ⃗i and ⃗j is not a unit vector because its magnitude is not unit. Yes, this sum can be multiplied by a scalar number to get a unit vector.
Explanation: Vectors ⃗i and ⃗j represent vectors of unit magnitudes along X-axis and Y-axis, so the angle between them is a right angle. Hence the magnitude of their vector sum will be √2. If we multiply this sum by a scalar number 1/√2 we can get a unit vector.
11. Let A=3i+4j. Write four vectors B such that A=/=B but A=B.
Answer: Four such vectors may be B=3i-4j, B=-3i+4j, B=-3i-4j and B= 5j
12. Can you have AxB=A.B with A=/=0 and B=/=0? What if one of the two vectors is zero?
Answer: If A=/=0 and B=/=0 then AxB cannot be equal to A.B because cross product is a vector quantity having a magnitude equal to ABsinθ and direction perpendicular to the plane containing A and B while the dot product A.B is a scalar quantity having its value equal to ABcosθ.
If one of the two vectors is zero then A.B=0 and magnitude of AxB is also zero. In this case, the direction of the zero vector has no significance.
13. If AxB=0, can you say that (a) A=B ,(b) A=/=B?
Answer: If AxB =0 then we can not say that (a) A=B or (b) A=/=B. Since AxB =ABsinθ, then if any of A, B or θ is zero then AxB =0 even if A=B or A=/=B.
14. If A=5i-4j and B= -7.5i+6j . Do we have B=kA? Can we say that B/ A=k?
Answer: In this case, we do have B=kA where k= -1.5. But we can not say that B/A=k because the division of vectors is not defined.
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CHAPTER- 18 - Geometrical Optics
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CHAPTER- 14 - Fluid Mechanics
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EXERCISES -Q-31 TO Q-40
EXERCISES -Q-41 TO Q-50
EXERCISES -Q-51 TO Q-60
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CHAPTER- 15 - Wave Motion and Waves on a String
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EXERCISES Q-1 TO Q-10
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CHAPTER- 18 - Geometrical Optics
CHAPTER- 16 - Sound Waves
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EXERCISES - Q-81 TO Q-89
CHAPTER- 15 - Wave Motion and Waves on a String
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EXERCISES- Q-1 TO Q-10
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CHAPTER- 13 - Fluid Mechanics
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EXERCISES Q-1 TO Q-10
EXERCISES- Q11 TO Q20
EXERCISES Q-21 TO Q30
EXERCISES Q-31 TO Q35
CHAPTER- 12 - Simple Harmonic Motion
EXERCISES- Q11 TO Q20
EXERCISES- Q21 TO Q30
EXERCISES- Q31 TO Q40
EXERCISES- Q41 TO Q50
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CHAPTER- 11 - Gravitation
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CHAPTER- 7 - Circular Motion
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CHAPTER- 6 - Friction
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CHAPTER- 6 - Friction
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CHAPTER- 5 - Newton's Laws of Motion
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CHAPTER- 4 - The Forces
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CHAPTER- 3 - Kinematics - Rest and Motion
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CHAPTER- 2 - "Vector related Problems"
CHAPTER- 2 - "Vector related Problems"
please explain question 9
ReplyDeleteI have added a diagram that will make it clear. Hope it helps!
DeleteThank you so much sir for making this account!
ReplyDeleteSir ,,i thick some solution is not correct for example chapter..force
ReplyDeleteShort answer question no 1
Thank u so very much
ReplyDeleteThank u sir
ReplyDeletethanks alottt !!!!!!!!!!! really helped
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