Answers to Problems on (Physics, and Mathematics) HC Verma's Questions for Short Answer (1-7)

 Q#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.

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.

Q#1
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.

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 120⁰  to others. In this situation, their resultant will be zero.

Q#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.

Q#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.

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.

Q#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.

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.

Q#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.

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.

Q#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.

Multiplication of vectors having different dimensions is common. Either as dot product or cross product.