CHAPTER 10

VECTORS POINTS TO REMEMBER 

A quantity that has magnitude as well as direction is called a vector. It is denoted by a directed line segment.



Two or more vectors which are parallel to same line are called collinear vectors.



Position vector of a point P(a, b, c) w.r.t. origin (0, 0, 0) is denoted by       OP , where OP  ai  b j  c k and OP  a 2  b 2  c 2 .



If A(x1, y1, z1) and B(x2, y2, z2) be any two points in space, then     AB   x 2  x 1  i   y 2  y 1  j   z 2  z 1 k and

 AB 

x 2

 x 1



2

 y 2  y 1

2

2

 z 2  z 1 .





If two vectors a and b are represented in magnitude and direction by   the two sides of a triangle taken in order, then their sum a  b is represented in magnitude and direction by third side of triangle taken in opposite order. This is called triangle law of addition of vectors.



If a is any vector and  is a scalar, then  a is a vector collinear with







 a and  a  







a .







If a and b are two collinear vectors, then a   b where  is some scalar. 





 



Any vector a can be written as a  a a , where a is a unit vector in 

the direction of a .

XII – Maths

101









If a and b be the position vectors of points A and B, and C is any point   which divides AB in ratio m : n internally then position vector c of point     mb  na . If C divides AB in ratio m : n externally, C is given as C  m  n    mb  na . then C  m – n

 The angles ,  and  made by r  ai  b j  ck with positive direction of x, y and z-axis are called direction angles and cosines of these angles  are called direction cosines of r usually denoted as l = cos , m = cos , n = cos . a b c Also l   , m   , n   and l2 + m2 + n2 = 1. r r r

 

  





The numbers a, b, c proportional to l, m, n are called direction ratios.     Scalar product of two vectors a and b is denoted as a.b and is defined       as a.b  a b cos , where  is the angle between a and b (0  ).     Dot product of two vectors is commutative i.e. a  b  b  a.

      a  b  0  a  o, b  o or    2 a  a  a , so i  l  j  j 

  a  b. k  k  1.

  If a  a 1i  a 2 j  a 3 k and b  b 1l  b 2 j  b 3 k, then   a  b = a1a2 + b1b2 + c1c2.     a . b  Projection of a on b  and projection vector of b

      a . b   a along b     b. b   

  Cross product or vector product of two vectors a and b is denoted as      a  b and is defined as a  b  a b sin  n . were  is the angle 102

XII – Maths

  between a and b (0 ) and n is a unit vector perpendicular to both     a and b such that a , b and n form a right handed system. 

    Cross product of two vectors is not commutative i.e., a × b  b × a ,     but a × b    b × a  .



         a × b  o  a = o , b = o or a || b .



 i  i  j  j  k  k  o .



i  j  k, j  k  i, k  i  j and j  i  –k, k  j  i, i  k   j



  If a  a1i  a2 j  a3 k and b  b1i  b2 j  b3 k , then i   a  b  a1 b1









k a3 b3

    a  b    Unit vector perpendicular to both a and b       .  a  b    a  b is the area of parallelogram whose adjacent sides are   a and b . 1     a  b is the area of parallelogram where diagonals are a and b . 2    If a , b and c forms a triangle, then area of the triangle. 



j a2 b2

  1   1  1  a  b  b  c = c  a . 2 2 2

   Scalar triple product of three vectors a , b and c is defined as     a .  b × c  and is denoted as  a b c 

XII – Maths

103



 Geometrically, absolute value of scalar triple product  a b c  represents    volume of a parallelepiped whose coterminous edges are a , b and c .



      a , b and c are coplanar   a b c   0



          a b c    b c a    c a b 



If

  ^ ^ ^ ^ ^ ^ a  a1 i  a2 j  a3 k , b  b1 i  b2 j  b3 k &  ^ ^ ^ c  c1 i  c 2 j  c 3 k , then

a1   a b c   b1 c1 

a2 b2 c2

a3 b3 c3

The scalar triple product of three vectors is zero if any two of them are same or collinear.

VERY SHORT ANSWER TYPE QUESTIONS (1 MARK) 1.

What are the horizontal and vertical components of a vector

 a of

magnitude 5 making an angle of 150° with the direction of x-axis. 2.

  What is a  R such that a x  1, where x  i  2 j  2k ?

3.

When is

4.

What is the area of a parallelogram whose sides are given by 2i – j and i  5k ?

    x  y  x  y ?

5.

  What is the angle between a and b , If     a  b  3 and a  b  3 3.

6.

Write a unit vector which makes an angle of

7.

  with x-axis and with 4 3

z-axis and an acute angle with y-axis.  If A is the point (4, 5) and vector AB has components 2 and 6 along x-axis and y-axis respectively then write point B. 104

XII – Maths

8.

9.

10.

11. 12.

What is the point of trisection of PQ nearer to P if positions of P and Q are 3i  3 j – 4k and 9i  8 j  10k respectively? Write the vector in the direction of 2i  3 j  2 3 k , whose magnitude is 10 units. What are the direction cosines of a vector equiangular with co-ordinate axes? What is the angle which the vector 3i – 6 j  2k makes with the x-axis? Write a unit vector perpendicular to both the vectors 3i – 2 j  k and – 2i  j – 2k .

13.

What is the projection of the vector i – j on the vector i  j ?

14.

If

15.

For what value of ,  b  2i  6 j  3k ?

16.

What is

17. 18.

      a  2, b  2 3 and a  b , what is the value of a  b ?  a  i  j  4k

is perpendicular to

       a , if  a  b  .  a – b   3 and 2 b  a ?

      What is the angle between a and b , if a – b  a  b ?   In a parallelogram ABCD, AB  2i  j  4k and AC  i  j  4k. What is the length of side BC ?

19.

20. 21.

22.

23.

What is the area of a parallelogram whose diagonals are given by vectors 2i  j  2k and i  2k ?    Find x if for a unit vector a , x – a . x  a  12 .







    If a and b are two unit vectors and a  b is also a unit vector   then what is the angle between a and b ? If i, j , k are the usual three mutually perpendicular unit vectors then  what is the value of i .  j  k   j .  i  k   k .  j  i  ?

      What is the angle between x and y if x . y  x  y ?

XII – Maths

105

24.

25.

26.

Write a unit vector in xy-plane, making an angle of 30° with the +ve direction of x–axis.        If a , b and c are unit vectors with a  b  c  0 , then what       is the value of a . b  b . c  c . a ?

    If a and b are unit vectors such that  a  2 b  is perpendicular     to  5 a  4 b  , then what is the angle between a and b ?

SHORT ANSWER TYPE QUESTIONS (4 MARKS) 27.

If ABCDEF is a regular hexagon then using triangle law of addition prove that :

       AB  AC  AD  AE  AF  3 AD  6 AO O being the centre of hexagon. 28.

Points L, M, N divides the sides BC, CA, AB of a ABC in the ratios    1 : 4, 3 : 2, 3 : 7 respectively. Prove that AL  BM  CN is a vector  parallel to CK where K divides AB in ratio 1 : 3.

29.

The scalar product of vector i  j  k with a unit vector along the sum of the vectors 2i  4 j – 5k and i  2 j  3k is equal to 1. Find the

30.

value of .    a , b and c are three mutually perpendicular vectors of equal    magnitude. Show that a  b + c makes equal angles with

31.

   –1  1  a , b and c with each angle as cos  .  3     If   3i  j and   2i  j  3k then express  in the form of          1   2 , where  1 is parallel to  and  2 is perpendicular  to  .

32.

  If a , b ,  that a 

     c are three vectors such that a  b  c  0 then prove      b  b  c  c  a.

106

XII – Maths

33.

34.

35.

36.

       a  3, b  5, c  7 and a  b  c  0 , find the angle   between a and b . If

      Let a  i  j , b  3 j – k and c  7i – k , find a vector d which     is perpendicular to a and b and c . d  1.   If a  i  j  k , c  j – k are the given vectors then find a vector       b satisfying the equation a  b  c , a . b  3. Find a unit vector perpendicular to plane ABC, when position vectors of

37.

i  j  3k and   For any two vector, show that a  b 

38.

Evaluate

39.

If a and b are unit vector inclined at an angle  than prove that :

A, B, C are

(i)

40. 41.

42. 43.

44.



a

sin

3i – j  2k ,

 i



2   a

 j



2   a

 1   .  2 2 a b

(ii)

For any two vectors, show that

 k

4i  3 j  k respectively.   a  b .

2 .

tan

  a  b 

 a   2 a  2

a b

2

b . b





a

 2  b  .

  ^   a  i  j  k , b  i  j  2 j and c  xi   x  2  j  k^ . If c   lies in the plane of a and b , then find the value of x. Prove that angle between any two diagonals of a cube is cos

1

 1 .  3

Let a, b and c are unit vectors such that  a· b   a· c  0 and the    c . , then prove that  angle between b and c is a   2 b 6 Prove that the normal vector to the plane containing three points with    position vectors a , b and c lies in the direction of vector       b  c  c  a  a  b.

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107

45.

46.

47.

   a , b , c are position vectors of the vertices A, B, C of a triangle ABC then show that the area of      1  ABC is a  b  b  c  c  a . 2         If a  b  c  d and a  c  b  d , then prove that         a  d is parallel to b – c provided a  d and b  c . If

Dot product of a vector with vectors i  j  3k, i  3j  2k

and 2i  j  4k is 0, 5 and 8 respectively. Find the vectors. 48.

49.

50.

51.

52. 53.

54.

55.

56.

   If a  5i  j  7k, b  i  j  k, find  such that a  b and   a  b are orthogonal.       Let a and b be vectors such that a  b  a  b  1,   then find a  b .

    If a  2, b  5 and a  b  2i  j  2kˆ, find the value of   a  b.       are three vectors such that a, b, c b  c  a and       a  b  c . Prove that a , b and c are mutually perpendicular to    each other and b  1, c  a .     If a  2iˆ  3 jˆ, b  iˆ  jˆ  kˆ and c  3iˆ  kˆ find  a b c  . Find volume of parallelepiped whose coterminous edges are given by    vectors a  2iˆ  3 jˆ  4kˆ, b  iˆ  2 jˆ  kˆ, and c  3iˆ  jˆ  2kˆ.

  Find the value of  such that a  iˆ  jˆ  kˆ, b  2iˆ  jˆ  kˆ and  c  iˆ  jˆ  kˆ are coplanar. Show that the four points (–1, 4, –3), (3, 2, –5) (–3, 8, –5) and (–3, 2, 1) are coplanar.

   For any three vectors a , b and c , prove that 108

XII – Maths

   a  b

57.

  b  c

     c  a   2  a b c         For any three vectors a , b and c , prove that a  b , b  c   and c  a are coplanar.

ANSWERS 5 3

,

5



3.

  x and y are like parallel vectors.

2

1 3

2. a  

.

1.

2

 3

4.

126 sq units.

5.

6.

1  1 1 i  j  k 2 2 2

7. (6, 11)

8.

10.

12.

 14   5, 3 , – 6  1



3

, 

1 3

3i  4 j  k 26

, 

9.

1 3

.

11.

4 i  6 j  4 3  k.

cos

1

 3 .  7

13. 0

.

14.

4

15. –9

16.

2

17.

 . 2

19.

3 sq. units. 2

18.

5

XII – Maths

109

20.

22.

13

–1

3 1 i  j 2 2

24.

21.

2 3

23.

 4

25.



3 2

26.

 3

29.

 = 1

31.

 3 1 1 3    i  j    i  j  3k  . 2    2 2 2

33.

60°

34.

35.

5 2 2 i  j  k. 3 3 3

36.

38.

 2 a

41. x = – 2

47.

i  2 j  k

49.

2

3

52.

4

54.

 = 1

1 1 3 i  j  k. 4 4 4 1 165

48.

 73

50.

91 10

10i  7 j  4k  .

53. 37

110

XII – Maths