VECTORS A . Components of vector in three dimensions : If P( x , y , z ) is a point in space and i , j , k are unit vectors , then

(i) op = xi + yj + zk (ii) Magnitude |OP | = √x2 + y2 + z2

(ii) Direction ratio ( d.r.)of a = xi + yj + zk is ( x , y , z )

and its direction cosine(d.c.) is ( 𝑥

√x2 + y2 +z2 ,

𝑦

√x2 + y2 +z2 ,

𝑧

√x2 + y2 +z2 )

(iii) Vector joining two points : If P( x1 , y1 , z1 ) and Q ( x2 , y2 , z2 ) are two points , then vector joining P and Q is

given by PQ = ( x2 − x1) i + ( y2 − y1)j + ( z2 − z1)k

B. (i) Unit vector : |a | = 1 ; a =a

| a |

(iii) Vector direction of a which has magnitude k units =k a = k(a

| a |)

C. Collinear or (parallel vector) (i) a ‖ b a = b

(ii) a = a1i + b1j + c1k ; b = a2i + b2j + c2k a1

a2=

b1

b2=

c1

c2

D. Section Formula : The position vector of a point R dividing the line segment joining the points P and Q , whose

position vectors are 𝑎 𝑎𝑛𝑑 ��

(i) in the ratio m : n internally is m b +n a

m +n (ii) in the ratio m : n externally is

m b − n a

m − n

(iii) mid – point is a + b

2

4. Projection of 𝑎 along 𝑏 is a .b

|b |

6. Product of two vectors (Scalar product or dot product ) : 𝑎 . 𝑏 = | 𝑎 || 𝑏 |𝑐𝑜𝑠𝜃 , where 0

Properties : (i) 𝑎 . 𝑏 is a real number. (ii) a . b = 0 a b (iii) i . i = j. j = k. k = 1, (iv) i . j = j. k = k. i = 0

(v) 𝑎 . 𝑏 = 𝑏 . 𝑎 (vi) 𝑎 . ( 𝑏 + 𝑐 ) = 𝑎 . 𝑏 + 𝑎 . 𝑐

7. Vector product ( cross product ) of two vectors : 𝑎 × 𝑏 = | 𝑎 || 𝑏 |𝑠𝑖𝑛𝜃 ��

Properties : (i) 𝑎 × 𝑏 is a vector . (ii) a × b = 0 a b (iii) i × i = j × j = k × k = 0,

(iv) i × j = k, j × k = i , k × i = j (v) 𝑎 × 𝑎 = 0 (vi) 𝑎 × ( 𝑏 + 𝑐 ) = 𝑎 . 𝑏 𝑎 . 𝑐

(v) 𝑎 × 𝑏 = − 𝑏 × 𝑎

8. (i) Area of the triangle ABC = 1

2 |𝑎 × 𝑏 | (ii) Area of the parallelogram(sides given) = |𝑎 × 𝑏 |

(iii) Area of the parallelogram ( diagonals given) = 1

2 |𝑑 1 × 𝑑2 |

9. If is the angle between a = a1i + b1j + c1k and a = a2i + b2j + c2k , cos = a1a2+b1b2+ c1c2

√ a12 + b1

2 + c1 2 √ a2

2 + b22 + c2

2

10. resultant of two vectors 𝑎 𝑎𝑛𝑑 𝑏 = 𝑎 + 𝑏

VECTORS 1. (i) Show that the vectors 2iˆ 3 ˆj 4kˆ and 4iˆ 6 ˆj 8kˆ are collinear.

(ii) Write the direction ratio’s of the vector a iˆ ˆj 2kˆ and hence calculate its direction cosines.

[Ans : ( 1, 1 , 2 ) , 1

√6𝑖 +

1

√6𝑗 −

2

√6 �� ]

2. Find unit vector in the direction of vector a 2iˆ 3 ˆj kˆ [ Ans : 2

√14𝑖 +

3

√14𝑗 +

1

√14 ��]

3. Find a vector in the direction of vector a iˆ 2 ˆj that has magnitude 7 units. [Ans : 7

√5𝑖 −

14

√5 𝑗 ]

4. Find a vector in the direction of vector 5iˆ ˆj 2kˆ which has magnitude 8 units.

[Ans : 40

√30𝑖 −

8

√30𝑗 +

16

√30 ��]

5. If a = i + 2j − k and b = 3i + j − 5kfind a unit vector in the direction of (i) 2a + b (ii) 3a − 4b

6. Consider two points P and Q with position vectors OP 3a 2b and OQ a b . Find the position

vector of a point R which divides the line joining P and Q in the ratio 2:1, (i) internally, and (ii)

externally. [Ans : 5

3a , 4a − b ]

7. Show that the points A(2iˆ ˆj kˆ), B(iˆ 3 ˆj 5kˆ), C(3iˆ 4 j 4kˆ) are the vertices of a right angled

triangle.

8. If a = 3i − j − 4k , b = −2i + j − 3k and c = i + 2j − k find a unit vector parallel to 3a − b + 4c

9. Find the unit vector in the direction of the sum of the vectors, a 2iˆ 2 ˆj – 5kˆ and b 2iˆ ˆj 3kˆ

[Ans : 4

√29𝑖 +

3

√29𝑗 −

2

√29 ��]

10. Find the values of x, y and z so that the vectors a xiˆ 2 ˆj zkˆ and b 2iˆ yˆj kˆ are equal.

[ Ans : x = 2, y = 2, z = 1]

11. Show that the vector iˆ ˆj kˆ is equally inclined to the axes OX, OY and OZ.

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

[ Ans : 1/3 ,2/3 , 2/3 ]

13. Find the position vector of a point R which divides the line joining two points P and Q whose position

vectors are iˆ 2 ˆj kˆ and – iˆ ˆj kˆ respectively, in the ratio 2 : 1 (i) internally (ii) externally

[Ans: ( 1/3 ,4/3 ,1/3) , ( 3 , 3 )]

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

[Ans (3,2,1)]

15. Show that the points A, B and C with position vectors, a 3iˆ 4 ˆj 4kˆ, b 2iˆ ˆj kˆ and c iˆ 3

ˆj 5kˆ , respectively form the vertices of a right angled triangle.

16. Find the angle between two vectors 𝑎 𝑎𝑛𝑑 𝑏 with magnitudes 1 and 2 respectively and when 𝑎 . 𝑏 1

[ Ans :/3]

17. Find angle between the vectors a iˆ ˆj kˆ and b iˆ ˆj kˆ . [ Ans : 𝑐𝑜𝑠−1(−1

3)]

18. If a 5iˆ ˆj 3kˆ and b iˆ 3 ˆj 5kˆ , then show that the vectors a b and a b are

perpendicular.

19. Find the projection of the vector a 2iˆ 3 ˆj 2kˆ on the vector b iˆ 2 ˆj kˆ [Ans :5

3 √6]

20. Find | a b | , if two vectors a and b are such that | a | 2, | b | 3 and a . b 4 [ Ans : √5]

21. If a is a unit vector and (x a ) (x a ) 8 , then find | x | . [ Ans : |x | = 3]

22. Prove that, For any two vectors a and b , we always have | a . b | | a | | b |

23. Prove that ,For any two vectors a and b , we always have | a b | | a | | b | (triangle inequality).

24. Show that the points A(2iˆ 3 ˆj 5kˆ), B( iˆ 2 ˆj 3kˆ) and C(7iˆ kˆ) are collinear.

25. Find | a | and | b | , if (a b ) (a b ) 8 and | a |8 | b | . [ Ans : 16√2

3√7 ,

2√2

3√7]

26. Evaluate the product (3a 5b ) (2a 7b ) . [Ans : 6 | a |2 +11a . b 35| b |2]

27. Find the magnitude of two vectors a and b , having the same magnitude and such that the angle between

them is 60o and their scalar product is ½ [Ans : | a | = 1 , | b | =1 ]

28 Find | x | , if for a unit vector a , (x a ).( x a ) 12 . [Ans : √3]

29. If a 2iˆ 2 ˆj 3kˆ, b iˆ 2 ˆj kˆ and c 3iˆ ˆj are such that a + b is perpendicular to c , then find the value of ns : 8

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ABC.

[ABC is the angle between the vectors AB and BC ] [ Ans :𝑐𝑜𝑠−1 10

√102 ]

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

32. Show that the vectors 2iˆ ˆj kˆ, iˆ 3 ˆj 5kˆ and 3iˆ 4 ˆj 4kˆ form the vertices of a right angled

triangle.

33. If a is a nonzero vector of magnitude ‘a’ and a nonzero scalar, then a is unit vector .Find

[Ans :1

| |]

34. Find | a b |, if a 2iˆ ˆj 3kˆ and b 3iˆ 5 ˆj 2kˆ [ Ans :√507]

35. Find a unit vector perpendicular to each of the vectors (a b ) and (a b ), where a iˆ ˆj kˆ,

b iˆ 2 ˆj 3kˆ . [Ans : − 1

√6𝑖 +

2

√6𝑗 −

1

√6 �� ]

36. Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1) as its vertices.

[ Ans :1

2√21]

37. Find the area of a parallelogram whose adjacent sides are given by the vectors a 3iˆ ˆj 4kˆ and

b iˆ ˆj kˆ. [Ans :√42]

38. Find | a b |, if a iˆ 7 ˆj 7kˆ and b 3iˆ 2 ˆj 2kˆ . [Ans :19√2 ]

39. Find a unit vector perpendicular to each of the vector a b and a b , where a 3iˆ 2 ˆj 2kˆ

and b iˆ 2 ˆj 2kˆ . [Ans : 2

3𝑖 −

2

3𝑗 −

1

3 �� ]

40. If a unit vector a makes angles 𝜋

3with 𝑖,

𝜋

4with 𝑗 and an acute angle with �� , then findand hence, the

components of a . [Ans 𝜋

3 ,

1

2𝑖 +

1

√2𝑗 +

1

2 �� ]

41. Show that (a b ) ( a b ) = 2(a b ).

42. Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5). [ Ans :√61

2]

43. Find the area of the parallelogram whose adjacent sides are determined by the vectors a iˆ ˆj 3kˆ

and b 2iˆ 7 ˆj kˆ . [Ans :15√2 ]

44. Let the vectors a and b be such that |a | =3 and | b | =√2

3 , then a b is a unit vector,

Find the angle between a and b . [ Ans 𝜋

4 ]

45. If iˆ ˆj kˆ, 2iˆ 5 ˆj, 3iˆ 2 ˆj 3kˆ and iˆ 6 ˆj kˆ are the position vectors of points A, B, C and D

respectively, then find the angle between AB and CD . Deduce that AB and CD are collinear.

[Ans : ]

46. Let a , b and c be three vectors such that | a |3, | b |4, |c |5 and each one of them being

perpendicular to the sum of the other two, find | a b c | . [Ans : 5√2]

47. Three vectors a , b and c satisfy the condition a b c 0 . Evaluate the quantity

a b b c + c a if | a |1, | b |4 and | c |2 . [Ans : −21

2 ]

48. If with reference to the right handed system of mutually perpendicular unit vectors iˆ, ˆj and kˆ,

𝛼 3iˆ ˆj, 𝛽 2iˆ ˆj – 3kˆ , then express 𝛽 in the form 𝛽 𝛽 1 𝛽 2 , where 𝛽 1 is parallel to

𝛼 and 𝛽 2 is perpendicular to 𝛼 49. 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. [Ans : − 5

2𝑖 +

3√3

2𝑗 ]

50. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a 2iˆ 3 ˆj kˆ and

b iˆ 2 ˆj kˆ . [Ans : 3

2√10 𝑖 +

√10

2𝑗]

51. If a iˆ ˆj kˆ, b 2iˆ ˆj 3kˆ and c iˆ 2 ˆj kˆ , find a unit vector parallel to the vector

2 a – b 3 c . [Ans : 3

√22𝑖 −

3

√22𝑗 +

2

√22 ��]

52. 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. [Ans . 2 : 3]

53. Let a iˆ 4 ˆj 2kˆ, b 3iˆ 2 ˆj 7kˆ and c 2iˆ ˆj 4kˆ . Find a vector d which is

perpendicular to both a and b , and c d 15 . [Ans : 1

3 ( 160 𝑖 − 5 𝑗 − 70 �� ]

54. The scalar product of the vector iˆ ˆj kˆ with a unit vector along the sum of vectors 2iˆ 4 ˆj 5kˆ and

iˆ 2 ˆj 3kˆ is equal to one. Find the value of . [ Ans : = 1]

55. Prove that (a b ) ( a b ) | a |2 | b |2 , if and only if a , b are perpendicular, given

a 0 b 0 56. Find value of iˆ.( ˆj kˆ) ˆj (iˆ kˆ) kˆ (iˆ ˆj) . [Ans : 1 ]

57. If �� and �� are unit vectors and is the angle between them , Prove that 𝑠𝑖𝑛𝜃

2=

1

2 |𝑎 − �� |

58. If | a | = √26 , | b | = 7 and | a b | = 35 , Find a . b [ Ans : 7 ]

59. If | a | = 5 , | b | = 13 and | a b | = 25 , Find a . b [ Ans : 60 ]

60. If | a | = 2 , | b | = 5 and | a b | = 8 , Find a . b [ Ans : 6 ]

61. Prove that | a b |2 = |𝑎 |2 |𝑏 |2 ( 𝑎 . 𝑏 )2 = |𝑎 . 𝑎 𝑎 . 𝑏

𝑎 . 𝑏 𝑏 . 𝑏 |

62. Prove that | a b | = ( 𝑎 . 𝑏 ) tan , where is the angle between 𝑎 𝑎𝑛𝑑 𝑏 .

63. In any triangle ABC , Prove that cosC = 𝑎2+ 𝑏2− 𝑐2

2𝑎𝑏 with help of vectors.

64. Using vector method , Prove that in a triangle ABC , 𝑎

𝑠𝑖𝑛𝐴=

𝑏

𝑠𝑖𝑛𝐵=

𝑐

𝑠𝑖𝑛𝐶 , Where a , b , c are the lengths

of the sides opposite respectively to the angle A , B , C of triangle ABC.

65. Prove that a (b c ) + b (c a ) + c (a b ) = 0

66. If Prove that a + b c = 0 , Prove that a b b c = c a

Prove that (a b ) ( a b ) = 2 ( 𝑎 × 𝑏 )

68. If a b c d and a c = b d , Show that (a d ) is parallel to (b c ).

69. If a = 4i + 3j + k and b = i − 2k , then find | 2b a | . [ Ans : 6√14 ]

70. Let a iˆ ˆj , b 3 ˆj kˆ and c 7iˆ kˆ . Find a vector d which is perpendicular to both a

and b , and c d 1 . [Ans : 1

4 ( 𝑖 + 𝑗 + 3 �� ]

Scalar triple product

* (product of three vectors) of 𝑎 , 𝑏 𝑎𝑛𝑑 𝑐 is [𝑎 𝑏 𝑐 ] =𝑎 . (𝑏 × 𝑐 )

* Properties of scalar triple product :

(a) [𝑎 𝑎 𝑐 ] = 0 , [𝑐 𝑎 𝑐 ] = 0 , [𝑎 𝑏 𝑏 ] = 0

(b) [𝑎 𝑏 𝑐 ] = −[𝑎 𝑐 𝑏 ] = [𝑐 𝑎 𝑏 ] , [𝑎 𝑏 𝑐 ] = −[𝑏 𝑎 𝑐 ] = [𝑏 𝑐 𝑎 ] etc.

(c) If three vectors are coplanar then [𝑎 𝑏 𝑐 ] = 0

(d) Volume of parallelepiped = | [𝑎 𝑏 𝑐 ] |

(e) Four points A, B, C, D are coplanar if [AB , AC , AD ] = 0

1. Scalar triple product

i. Prove that [ 𝑖 �� �� ] = 1 and [ 𝑖 �� �� ] = −1

ii. If 𝑎 = 2𝑖 + 𝑗 + 3�� 𝑏 = − 𝑖 + 2𝑗 + �� and 𝑐 = − 3𝑖 + 𝑗 + 2�� , Find [𝑎 𝑏 𝑐 ] Ans:20

iii. Find the volume of the parallelepiped whose coterminous edges are represented by the vectors

𝑎 = 2𝑖 − 3𝑗 + �� , 𝑏 = 𝑖 − 𝑗 + 2�� and 𝑐 = 2𝑖 + 𝑗 − �� [Ans : |−14| = 14]

iv. Show that the vectors 𝑖 − 3𝑗 + 4�� , 2 𝑖 − 𝑗 + 2�� and 4�� − 7𝑗 + 10�� are coplanar.

v. Find the value of so that the vectors 𝑎 = 2𝑖 − 3𝑗 + �� , 𝑏 = 𝑖 + 2𝑗 − 3�� and

𝑐 = 𝑗 + �� are coplanar.

vi. Show that the four points with position vectors 4𝑖 + 5𝑗 + 3�� , − 𝑗 − �� , 3𝑖 + 9�� + 4��

and 4(−𝑖 + 𝑗 + ��) are coplanar.

vii. Prove that any three vectors 𝑎 , 𝑏 , 𝑐 [𝑎 + 𝑏 𝑏 + 𝑐 𝑐 + 𝑎 ] = 2[𝑎 𝑏 𝑐 ]

2. Find the value of , if four points with position vectors 3𝑖 + 6𝑗 + 9�� , 𝑖 + 2𝑗 + 3�� , 2𝑖 + 3𝑗 + ��

and 4𝑖 + 6𝑗 + �� are coplanar.

3. If the vector P = 𝑎𝑖 + 𝑗 + �� , q = 𝑖 + 𝑏𝑗 + �� and r = 𝑖 + 𝑗 + 𝑐�� are coplanar , then for a, b, c 0

Then show that 1

1−𝑎 +

1

1−𝑏+

1

1−𝑐= 1

4. Find x such that the four points A( 4 , 1 , 2) , B( 5 , x , 6) , C(5 , 1 , 1) and D( 7 , 4 ,0 ) are coplanar. [Ans: 4]

5. Prove that 𝑏 . { (𝑏 + 𝑐 ) × (𝑎 + 2𝑏 + 3 𝑐 )} = [ 𝑎 𝑏 𝑐 ]