Chapter 10 . Vector Algebra

EXERCISE 10.1

1. Represent graphically a displacement of 40 km, 30° east of north.

Solution :

        

This diagram represents a displacement of 40 km, 30° east of north.
2. Classify the following measures as scalars and vectors.
(i) 10 kg (ii) 2 meters north-west (iii) 40°
(iv) 40 watt (v) 10 –19 coulomb (vi) 20 m/s²

Solution: (i) 10 kg - Scalar (It only has magnitude, representing mass)

(ii) 2 meters north-west - Vector (It has both magnitude, 2 meters, and direction, north-west)

(iii) 40° - Scalar (It represents an angle, which is a magnitude without direction)

(iv) 40 watt - Scalar (It represents power, which is a scalar quantity)

(v) 10−1910−19 coulomb - Scalar (It represents electric charge, a scalar quantity)

(vi) 20 m/s² - Vector (It represents acceleration, which has both magnitude and direction)

[Note : Scalars are quantities that have only magnitude, while vectors are quantities that have both magnitude and direction.]

3. Classify the following as scalar and vector quantities.
(i) time period (ii) distance (iii) force (iv) velocity (v) work done

Solution: (i) Time period – Scalar  

It is the duration of a complete cycle of a periodic motion and is represented by a scalar value.

(ii) Distance – Scalar  

It is the measure of how much ground an object has covered during its motion. It only has magnitude.

(iii) Force – Vector  

It is a push or pull acting upon an object, characterized by both magnitude and direction.

(iv) Velocity – Vector  

It is the rate of change of displacement with respect to time and has both magnitude and direction.

(v) Work done – Scalar  

It is the product of force and the displacement of an object in the direction of the force applied. It is a scalar quantity.

4. In Fig 10.6 (a square), identify the following vectors.

  

                 Fig 10.6
(i) Coinitial (ii) Equal (iii) Collinear but not equal

Solution :  (i) Coinitial Vectors :  and .

(ii) Equal Vectors :  and  .

 (iii) Collinear but not equal Vectors :  and .

5. Answer the following as true or false.
(i)  and -  are collinear.
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.

Solution :  (i) True -  and  are collinear because they lie on the same line, even if they are in opposite directions.

(ii) False - Two collinear vectors may have different magnitudes; their collinearity implies they lie on the same line, but their magnitudes can vary.

(iii) False - Two vectors having the same magnitude are not necessarily collinear. They could be pointing in different directions and not lie on the same line.

(iv) False - While collinear vectors lie on the same line, even if they have the same magnitude, they may not necessarily be equal. Equal vectors have the same magnitude and direction. Collinear vectors can have the same or different magnitudes.

EXERCISE 10.2

1. Compute the magnitude of the following vectors:
     ,    , 

Solution :  We have,  

     

Again,   

  

and

2. Write two different vectors having same magnitude.

Solution : Let  and  

 

and

Therefore , the magnitudes of the vectors are the same .
3. Write two different vectors having same direction.
Solution: Let  and  

The direction cosines of the given vectors are : 

and    

 Again, the direction cosines of the given vectors are :
4. Find the values of  and  so that the vectors  and  are equal .

Solution :  let  and

A/Q,  

Comparing the coefficients of   and  , we get

       and

   

The value of and  are 2 and 3 respectively .

5. Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (– 5, 7).

Solution: Given, the initial point (2,1) and terminal point (−5,7) .

Scalar components  

So, the scalar components of the vector are (−7,6).

Vector components  .

So, the scalar components of the vector are −7 and 6 , and the vector components are and  .

6. Find the sum of the vectors  ,  and  .

Solution : Given,  ,  and  

7. Find the unit vector in the direction of the vector  .

Solution : We have,

Thus, the required unit vector is

8. Find the unit vector in the direction of vector   where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

Solution :  Given, P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

  

Now,  

The unit vector in the direction of  , then

9. For given vectors,  anda  , find the unit vector in the direction of the vector  .

Solution : Given,  and

Thus, the required unit vector of  is

10. Find a vector in the direction of vector  which has magnitude 8 units.

Solution : Let

The unit vector in the direction of the given vector  is

Therefore, the vector having magnitude equal to 8 and in the direction of  is

11. Show that the vectors  and  are collinear.

Solution : Let and

We have, 

, where 

So, the vectors and are same direction .

Therefore, the vectors  and  are collinear.
12. Find the direction cosines of the vector  .

Solution : Let ,

The unit vector in the direction of the given vector is

The direction cosine of the given vectors are : 

13. Find the direction cosines of the vector joining the points A (1, 2, –3) and B(–1, –2, 1), directed from A to B.

Solution :  Since the vector is to be directed from A to B , clearly P is the initial point and Q is the terminal point.

So, the required vector joining P and Q is the vector , given by

The unit vector in the direction of the given vector is

The direction cosine of the given vectors are :

14. Show that the vector  is equally inclined to the axes OX, OY and OZ.

Solution : Let

The direction cosines of the given vectors are :

Therefore, the vector  is equally inclined to the axes OX, OY and OZ .  

15. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  and  respectively, in the ratio 2 : 1
(i) internally (ii) externally .

Solution : The position vector of P and Q are :                                           

   and

(i)  Here,

Using Section formula , we have

(ii)  Here,

Using  Section formula , we have

16. Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, –2).

Solution : Given, the position vector of the point P(2, 3, 4) is    

 And the position vector of the point Q(4, 1, –2) is .

   

Therefore, the position vector of the mid-point of vector

 

17. Show that the points A, B and C with position vectors, ,   and   , respectively form the vertices of a right angled triangle.

Solution : Given, the position vectors are :

,   and  

We have,

   

 units  

Again,

 

 units

And

 units

Now, 

Hence, the triangle is a right angled triangle.

18. In triangle ABC (Fig 10.18), which of the following is not true:

      
(A)   .
(B) 
(C) 
(D) 

Solution : Using the triangle law of vector addition , we have

Option (C) is not true .
19. If  and  are two collinear vectors, then which of the following are incorrect:
(A)  , for some scalar  
(B)    .
(C) the respective components of  and  are proportional
(D) both the vectors  and  have same direction, but different magnitudes.

Solution : (A)  , for some scalar  :

If  and  are two collinear vectors, then  is a scalar multiple of . So, the statement is correct.

 (B)   :

If  and  are collinear, they could have the same direction (positive sign) or opposite direction (negative sign).  So, the statement is correct.

 (C) The respective components of  and  are proportional :

If  and   may be collinear, their components may not necessarily be proportional unless they have the same direction. So, the statement is incorrect.

(D) Both vectors  and  have the same direction but different magnitudes :

If  and  are collinear, they have the same or opposite direction and may have the same magnitude if they are scalar multiples of each other. So,the statement is incorrect.

EXERCISE 10.3

1. Find the angle between two vectors  and  with magnitudes  and 2 , respectively having  .

Solution : Here,  ,  and

Let the angle θ between two vectors  and  is given by

Therefore, the angle between two vectors  and  is .
2. Find the angle between the vectors  and  .

Solution:  Let  and

 and

Let the angle θ between two vectors  and  is given by

 

3. Find the projection of the vector  on the vector  .

Solution : Let   and .

  

The projection of vector  on the vector  is given by

4. Find the projection of the vector  on the vector  .

Solution:  Let  and

The projection of vector  on the vector  is given by

5. Show that each of the given three vectors is a unit vector: , ,
 Also, show that they are mutually perpendicular to each other.

Solution: Let

Now,

Therefore, the given three vector is a unit vector .

Thus , the given three vectors are mutually perpendicular to each other .

6. Find  and  , if  and  .

Solution : We have,

and

7. Evaluate the product  .

Solution : We have, 

8. Find the magnitude of two vectors  and  , having the same magnitude and such that the angle between them is 60° and their scalar product is   .

Solution : Given,   , and

We have,

 

9. Find  , if for a unit vector  ,  .

Solution:  Here, 

We have,

10. If   ,   and  are such that  is perpendicular to  , then find the value of  .

Solution: Given,  ,   and

We have,

Therefore, the value of  is 8 .

11. Show that  is perpendicular to   , for any two nonzero vectors  and  .

Solution:  We have,

Therefore,  is perpendicular to    .

12. If  and   , then what can be concluded about the vector  ?

Solution:  Since, 

 

It implies that the vector is the zero vector.

 And    

It means that  and are perpendicular vectors.

 So, the conclusion in this case is that is perpendicular to  , and  is the zero vector,  can be any vector .

13. If   are unit vectors such that  , find the value of  .

Solution: Given,   

We have, 

14. If either vector   or   , then   . But the converse need not be true. Justify your answer with an example.

Solution: If either vector   or   , then  . This is because the dot product of any vector with the zero vector is always zero.

However, the converse is not necessarily true. That means, even if , it doesn't imply that either   or  is the zero vector.

Example: Consider the vectors  and  .

So, ,but neither  nor  is the zero vector.

This example illustrates that even if the dot product is zero, it doesn't necessarily mean that one of the vectors is the zero vector.

15. If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find  . [   is the angle between the vectors  and  ].

Solution:  Since, the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively .

 

and

 

16. Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.

Solution : Given, the points are A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1)

  Now ,   

We have,

Therefore, the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.

17. Show that the vectors  , and  form the vertices of a right angled triangle.

Solution: Let , the position vectors are  ,  and

 

 

Therefore, the vectors  ,  and  form the vertices of a right angled triangle.

18. If  is a nonzero vector of magnitude ‘a’ and  a nonzero scalar, then  is unit vector if
(A)   (B)     (C)     (D)

Solution: The vector  is unit vector, then

Correct option : (D)

EXERCISE 10.4

1. Find  if   and .

Solution:  Here,  and  .

2. Find a unit vector perpendicular to each of the vector  and  , where  and  .

Solution:  Here,  and   .

Therefore, the required unit vector is 

3. If a unit vector  makes angles  with  ,  with  and an acute angle  with  , then find  and hence, the components of  .

Solution: Here,

We have,

Therefore, the components of  are :

i.e.,

4. Show that  .

Solution:  L.H.S :

  R.H.S  Proved.

5. Find  and  if .

Solution:  We have,     

        

    

     and

          

Therefore, the value of  and  .
6. Given that  and   . What can you conclude about the vectors  and  ?

Solution: If  and be two nonzero vectors .

Then  if and only if  and are perpendicular to each other.

 i.e.,   .

Then  if and only if  and are parallel (or collinear) to each other.

i.e.,   .

7. Let the vectors  be given as  ,  and  . Then show that  .

Solution :  Given, the vectors are :

 ,  and  

We have,

 

LHS : 

  

RHS : 

  

LHS = RHS  Proved .
8. If either  or  , then  . Is the converse true? Justify your answer with an example.

Solution: Let  and

 

 and

No, the converse is not necessarily true. That is, even if  , it doesn't necessarily mean that either  or .

9. Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

Solution : Given, the vertices of the triangles are A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).  

We have,

Area of triangle ABC

Square units .

10. Find the area of the parallelogram whose adjacent sides are determined by the vectors :  and

Solution: Given, the vectors are  and

 Now,

 square units .

Therefore, the area of the parallelogram is  square units .

11. Let the vectors  and  be such that  and  , then  is a unit vector, if the angle between  and  is 

    (A)        (B)         (C)         (D)  

Solution: Given,  , and

Let  the angle between  and  .

 

(B)           

12. Area of a rectangle having vertices A, B, C and D with position vectors , , and respectively is
      (A)       (B) 1      (C) 2     (D) 4

Solution:  We have,

Therefore, the area of rectangle ABCD is 2 sq. units .

(C) 2

Miscellaneous Exercise on Chapter 10

1. Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Solution :  A unit vector in the XY-plane making an angle of 30° with the positive direction of the x-axis .

 

Given,  that the angle between the vector and the positive x-axis is 30°, we have:

  and  

Let

Therefore, the unit vector of  is 1 .

2. Find the scalar components and magnitude of the vector joining the points P() and Q ( ) .

Solution :  Given, the vector joining the points P() and Q (  ) is given by

The scalar components are  ,  and  .

The magnitude of 

3. A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

Solution :  Let O and Q are the initial and final positions of the girl respectively . 

She walks 4 km towards the west, so her displacement along the x-axis is – 4 km, then

Then, she walks 3 km in a direction 30° east of north.

We have ,

Therefore, the girl's displacement from her initial point of departure is , in the direction north-east of her starting point.

4. If  , then is it true that  ? Justify your answer.

Solution :   No, it is not necessarily true that  .

The magnitude of a vector is a scalar quantity representing its length.

 Since,  , then represents the resultant vector obtained by adding vectors  and   .

However, the magnitude of the resultant vector  might not be equal to the sum of the magnitudes of   and . This is because vector addition involves both magnitude and direction.

5. Find the value of  for which   is a unit vector.

Solution :  Given,   is a unit vector.

Therefore, the value of are

6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors  and  .

Solution :  Given,  and  .
Let   

 

The unit vector in the direction of the given vector  is

7. If  ,  and  , find a unit vector parallel to the vector  .

Solution : Given, vectors are  ,  and

Let

 

Therefore, the required unit vector

8. Show that the points A (1, – 2, – 8), B (5, 0, – 2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Solution :  Given, the points A (1, – 2, – 8), B (5, 0, – 2) and C (11, 3, 7) respectively .

Hence , the points A, B and C are collinear .

Again,

Let the position vector of the point B which divides AC internally in the ratio , then

 

Comparing the coefficient of  and on both side, we get

 ,,

  ,

Again,

And 

 

Therefore, the ratio is 2 : 3 .

9. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  and  externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.

Solution :  Here,

  and 

The position vector of a point R which divides the line joining two points P and Q externally in the ratio 1 : 2

Using section formula , we get

Again, P is the mid point of the line segment RQ
10. The two adjacent sides of a parallelogram are  and  . Find the unit vector parallel to its diagonal. Also, find its area.

Solution : Let ,  , 

And 

 

The unit vector parallel to its diagonal

Square units

 Therefore, the area of the parallelogram is square units  .
11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are  .

Solution : Let  a vector that is equally inclined to the axes OX, OY, and OZ , then the direction cosines are  and.

We have ,

Hence, the direction cosines of a vector equally inclined to the axes OX, OY and OZ are 

12. Let  ,  and  . Find a vector  which is perpendicular to both  and   and  .

Solution :  Given,  ,  and

Let 

A/Q,

Therefore, the vector  is

13. The scalar product of the vector  with a unit vector along the sum of vectors  and  is equal to one. Find the value of    .

Solution :  Let  ,  and

Let,  

A/Q,   

Therefore, the value of  is 1 .

14. If  are mutually perpendicular vectors of equal magnitudes, show that the vector   is equally inclined to  ,  and  .

Solution :  Given,  and  

Let, the cosines of the angles between   and  ,  and  are andrespectively .

Now , 

 

Again,

And 

Since,

So,   and also  

Hence, the vector is equally inclined to a , b and .

15. Prove that , if and only if  are perpendicular, given   ,  .

Solution :  We have,

Therefore,  and are perpendicular .

Choose the correct answer in Exercises 16 to 19.
16. If  is the angle between two vectors  and  , then  only when
(A)         (B)       (C)      (D)

Solution :  Given, is the angle between two vectors  and .

 

For  to be greater than or equal to zero, the cosine of the angle  must be greater than or equal to zero, as the magnitudes  and are always positive. So ,

The correct Answer (B) :

17. Let  and  be two unit vectors and  is the angle between them. Then  is a unit vector if
(A)         (B)        (C)       (D)

Solution :  Given, and

And

Correct Answer : (B)

18. The value of   is
        (A) 0            (B) –1             (C) 1            (D) 3

Solution :  We have,

 

The correct Answer : (C) 1  .

19. If  is the angle between any two vectors  and , then  when  is equal to
(A) 0         (B)             (C)        (D)  

Solution :  We have,

The correct answer : (B)   .