Q: 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?

No, the vector sum of the unit vectors i and j is not a unit vector, because the magnitude of the resultant of i and j is not one. Magnitude of the resultant vector is given by R = 12+12+cos90∘=2Yes, we can multiply this resultant vector by a scalar number 21 to get a unit vector.

Question 1

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

Accepted Answer

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.

Question 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?

Accepted Answer

No, it is not possible to obtain zero by adding two vectors of unequal magnitudes.Example: Let us add two vectors  A and  B of unequal magnitudes acting in opposite directions. The resultant vector is given byR=A2+B2+2ABcosθ​.If two vectors are exactly opposite to each other, then θ=180∘,cos180∘=−1R=A2+B2−2AB​From the above equation, we can say that the resultant vector is zero (R = 0) when the magnitudes of the vectors  R=A2+B2−2AB​ and R=A2+B2−2AB​ 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 magnitudesR=A2+B2−2AB​ ,given these three vectors make an angle of R=A2+B2−2AB​ with each other. Consider the figure below:Lets examine the components of the three vectors. R=A2+B2−2AB​Ay​=0⇒R=(A−B) or (B−A)By​=Bsin60∘Ay​=0Cy​=−Csin60∘R=A2+B2−2AB​So, along the x - axis , we have: By​=Bsin60∘⇒Bsin60∘−Csin60∘=0Hence, proved.

Question 3

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

Accepted Answer

A zero vector has physical significance in physics, as the operations on the zero vector gives us a vector.For any vector A , assume thatA + 0 = A A − 0 = AA × 0 = 0Again, for any real number λ we have:λ0=0The 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.

Question 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?

Accepted Answer

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 i,^​−i^ and j^​ 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 ( i,^​−i^ and j^​) are the same, but their directions are different.So, the resultant of i^ and −i^ is a zero vector.Now, j^​+0=j^​    (Using the property of zero vector)∴ The resultant of three unit vectors ( i,^​−i^ and j^​)  is a unit vector (j^​).

Question 5

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

Accepted Answer

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.

Question 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.

Accepted Answer

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.

Question 7

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

Accepted Answer

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,τ=r×F

Question 8

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

Accepted Answer

Yes, a vector can have zero components along a line and still have a nonzero magnitude. Example: Consider a two dimensional vector 2i^+0j^​. 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 2i^+0j^​ =  22+02​=2

Question 9

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

Accepted Answer

The direction of - A is opposite to A. So, if vector A and −A 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 −A.

Question 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?

Accepted Answer

No, the vector sum of the unit vectors i and j​  is not a unit vector, because the magnitude of the resultant of i and j​ is not one. Magnitude of the resultant vector is given by R = 12+12+cos90∘​=2​Yes, we can multiply this resultant vector by a scalar number 2​1​ to get a unit vector.

Question 11

Let A=3i+4j​. Write a vector B such that A=B, but A = B.

Accepted Answer

A vector B such that  A=B, but A = B are as follows:(i)  B=3i−4j​(ii)  B=3j​+4k(iii)  B=3k+4i(iv) B=3j​−4k

Question 12

Can you have  A×B=A⋅B with A ≠ 0 and B ≠ 0 ? What if one of the two vectors is zero?

Accepted Answer

No, we cannot have A×B=A⋅B 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.

Question 13

If A×B=0 can you say that:(a) A=B,(b) A=B ?

Accepted Answer

If  A×B=0, then both the vectors are either parallel or antiparallel, i.e., the angle between the vectors is either 0∘ or 180∘.ABsin θ n^=0.......(∵sin0∘=sin180∘=0)Both the conditions can be satisfied:(a) A=B, i.e., the two vectors are equal in magnitude and parallel to each other(b) A=B,  i.e., the two vectors are unequal in magnitude and parallel or anti parallel to each other.

Question 14

Let A=5i−4j​ and B=−7⋅5i+6j​. Do we have B=kA ? Can we say AB​ = k ?

Accepted Answer

If A=5i−4j​ and B=−7⋅5i+6j​, then we have B=kA by putting the value of scalar k as −1.5 . However, we cannot say that AB​ = k, because a vector cannot be divided by other vectors, as vector division is not possible.

Question 15

A vector is not changed if:

Accepted Answer

(d) It is slid parallel to itself.A vector is defined by its magnitude and direction. If we slide it to a parallel position to itself, then none of the given parameters, which define the vector, will change.Let the magnitude of a displacement vector ( A )  directed towards the north be 5 metres. If we slide it parallel to itself, then the direction and magnitude will not change.

Question 16

Which of the sets given below may represent the magnitudes of three vectors adding to zero?

Accepted Answer

1,2 and 1 may represent the magnitudes of three vectors adding to zero. For example one of the vectors of length 1 should make an angle of  135∘ with x-axis and the other vector of length 1 makes an angle of 225∘  with x-axis. The third vector of length 2 should lie along x-axis.

Question 17

The resultant of  A and B makes an angle α with  A and β with B:

Accepted Answer

(c) α  B.The resultant of two vectors is closer to the vector with the greater magnitude.Thus, α  B.

Question 18

The component of a vector is:

Accepted Answer

All the given options are incorrect. The component of a vector may be less than, greater than or equal to its magnitude, depending upon the vector and its components. But not always.

Question 19

A vector A points vertically upward and B points towards the north. The vector product A×B is:

Accepted Answer

(a) Along the westThe vector product  A×B will point towards the west. We can determine this direction using the right hand thumb rule.

Question 20

The radius of a circle is stated as 2.12 cm. Its area should be written as:

Accepted Answer

(c) 14.1 cm2.Area of a circle, A=  πr2 On putting the values, we get:A=722​×2.12×2.12⇒A=14.1cm2The rules to determine the number of significant digits says that in the multiplication of two or more numbers, the number of significant digits in the answer should be equal to that of the number with the minimum number of significant digits. Here, 2.12 cm has a minimum of three significant digits. So, the answer must be written in three significant digits.

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