HC Verma Questions and Solutions: Chapter 2- Physics & Mathematics- 1 | HC Verma Solutions - JEE PDF Download

Q.1. Is a vector necessarily changed if it is rotated through an angle?

Yes. A vector is defined by its magnitude and direction, so a vector can be changed by changing its magnitude and direction. If we rotate it through an angle, its direction changes and we can say that the vector has changed.

Q.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?

No, it is not possible to obtain zero by adding two vectors of unequal magnitudes.
Example: Let us add two vectors  of unequal magnitudes acting in opposite directions. The resultant vector is given by

If two vectors are exactly opposite to each other, then

From the above equation, we can say that the resultant vector is zero (R = 0) when the magnitudes of the vectors   are equal (A = B) and both are acting in the opposite directions.
Yes, it is possible to add three vectors of equal magnitudes and get zero.
Lets take three vectors of equal magnitudes  given these three vectors make an angle of 120° with each other. Consider the figure below:

Lets examine the components of the three vectors.

Here, A = B = C
So, along the x - axis , we have:

Hence, proved.

Q.3. Does the phrase "direction of zero vector" have physical significance? Discuss it terms of velocity, force etc.

A zero vector has physical significance in physics, as the operations on the zero vector gives us a vector.

For any vector  assume that



Again, for any real number λ we have:
 
The significance of a zero vector can be better understood through the following examples:
The displacement vector of a stationary body for a time interval is a zero vector.
Similarly, the velocity vector of the stationary body is a zero vector.
When a ball, thrown upward from the ground, falls to the ground, the displacement vector is a zero vector, which defines the displacement of the ball.

Q.4. Can you add three unit vectors to get a unit vector? Does your answer change if two unit vectors are along the coordinate axes?

Yes we can add three unit vectors to get a unit vector.
No, the answer does not change if two unit vectors are along the coordinate axes. Assume three unit vectors  along the positive x-axis, negative x-axis and positive y-axis, respectively. Consider the figure given below:

The magnitudes of the three unit vectors  are the same, but their directions are different.
So, the resultant of  is a zero vector.
Now,  (Using the property of zero vector)
∴ The resultant of three unit vectors  is a unit vector

Q.5. Can we have physical quantities having magnitude and direction which are not vectors?

Yes, there are physical quantities like electric current and pressure which have magnitudes and directions, but are not considered as vectors because they do not follow vector laws of addition.

Q.6. Which of the following two statements is more appropriate?
(a) Two forces are added using triangle rule because force is a vector quantity.
(b) Force is a vector quantity because two forces are added using triangle rule.

Two forces are added using triangle rule, because force is a vector quantity. This statement is more appropriate, because we know that force is a vector quantity and only vectors are added using triangle rule.

Q.7.  Can you add two vectors representing physical quantities having different dimensions? Can you multiply two vectors representing physical quantities having different dimensions?

No, we cannot add two vectors representing physical quantities of different dimensions. However, we can multiply two vectors representing physical quantities with different dimensions.
Example: Torque,

Q.8. Can a vector have zero component along a line and still have nonzero magnitude?

Yes, a vector can have zero components along a line and still have a nonzero magnitude.
Example: Consider a two dimensional vector  This vector has zero components along a line lying along the Y-axis and a nonzero component along the X-axis. The magnitude of the vector is also nonzero. Now, magnitude of

Q.9. Let ε₁ and ε₂ be the angles made by  with the positive X-axis. Show that tan ε₁ = tan ε₂. Thus, giving tan ε does not uniquely determine the direction of

The direction of -  is opposite to  So, if vector  make the angles ε₁ and ε₂ with the X-axis, respectively, then ε₁ is equal to ε₂ as shown in the figure:

Here, tan ε₁ = tan ε₂
Because these are alternate angles.
Thus, giving tan ε does not uniquely determine the direction of

Q.10. Is the vector sum of the unit vectors  a unit vector? If no, can you multiply this sum by a scalar number to get a unit vector?

No, the vector sum of the unit vectors  is not a unit vector, because the magnitude of the resultant of  is not one.
Magnitude of the resultant vector is given by
Yes, we can multiply this resultant vector by a scalar number  to get a unit vector.

Q.11. Let  Write a vector  such that  but A = B.

A vector  such that  but A = B are as follows:

Q.12. Can you have   with A ≠ 0 and B ≠ 0 ? What if one of the two vectors is zero?

No, we cannot have  with A ≠ 0 and B ≠ 0. This is because the left hand side of the given equation gives a vector quantity, while the right hand side gives a scalar quantity. However, if one of the two vectors is zero, then both the sides will be equal to zero and the relation will be valid.

Q.13. If  can you say that

If  then both the vectors are either parallel or antiparallel, i.e., the angle between the vectors is either  

Both the conditions can be satisfied:
(a)  i.e., the two vectors are equal in magnitude and parallel to each other
(b)  i.e., the two vectors are unequal in magnitude and parallel or anti parallel to each other.

Q.14. Let  Do we have  Can we say

If  then we have  by putting the value of scalar k as  
However, we cannot say that  because a vector cannot be divided by other vectors, as vector division is not possible.

Question for HC Verma Questions and Solutions: Chapter 2- Physics & Mathematics- 1

Try yourself:The resultant of  makes an angle α with  and β with 

• A.

  α < β
• B.

  α < β if A < B
• C.

  α < β if A > B
• D.

  α < β if A = B

Explanation

The resultant of two vectors is closer to the vector with the greater magnitude. Thus, α < β if A > B

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Question for HC Verma Questions and Solutions: Chapter 2- Physics & Mathematics- 1

Try yourself:A vector  points vertically upward and  points towards the north. The vector product  is

• A.

  along the west
• B.

  along the east
• C.

  zero
• D.

  vertically downward.

Explanation

The vector product  will point towards the west. We can determine this direction using the right hand thumb rule.

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Question for HC Verma Questions and Solutions: Chapter 2- Physics & Mathematics- 1

Try yourself:Let 

• A.

   is always greater than 
• B.

  It is possible to have 
• C.

  C is always equal to A + B
• D.

  C is never equal to A + B.

Explanation

It is possible to have 
Statements (a), (c) and (d) are incorrect.

Here, the magnitude of the resultant vector may or may not be equal to or less than the magnitudes of  or the sum of the magnitudes of both the vectors if the two vectors are in opposite directions.

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Question for HC Verma Questions and Solutions: Chapter 2- Physics & Mathematics- 1

Try yourself:Let the angle between two nonzero vectors  be 120° and its resultant be 

• A.

  C must be equal to 
• B.

  C must be less than 
• C.

  C must be greater than 
• D.

  C may be equal to 

Explanation

Here, we have three vector A, B and C.

Subtracting (i) from (ii), we get:

Using the resultant property 

Since cosine is negative in the second quadrant, C must be less than 

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Question for HC Verma Questions and Solutions: Chapter 2- Physics & Mathematics- 1

Try yourself:The magnitude of the vector product of two vectors  may be

• A.

  greater than AB
• B.

  equal to AB
• C.

  less than AB
• D.

  equal to zero.

Explanation

(b) equal to AB
(c) less than AB
(d) equal to zero.
The magnitude of the vector product of two vectors  may be less than or equal to AB, or equal to zero, but cannot be greater than AB.

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