GRADE 12 VELAMMAL VIDHYASHRAM, MAMBAKKAM

QUESTION BANK – II - MATHEMATICS

TOPIC: UNIT X - VECTORS
TWO MARKS TYPE
1. Find the unit vector in the direction of a⃗ =i+2 ^ ^j−3 k^
2. Find the unit vector in the direction of a⃗ =i−2 ^ ^j+ k^ which has magnitude of 7 units.
3. For the given vectors a⃗ =2 i− ^ ^j+2 k^ ∧⃗b=−i− ^ ^j+ k^ , find the unit vector ∈the direction of ⃗a + ⃗b
^
4. Find the unit vector in the direction of ⃗ PQ where P and Q are (1,2,3) and Q (4,5,6)
5. Find the value of x for which x(i−2 ^ ^j+ k^ ¿ is a unit vector.
6. Find a vector of magnitude 5 units parallel to the resultant of the vectors a⃗ =i−2 ^ ^j and b=3
⃗ ^ 2 ^j−k^
i+
7. Find a vector of magnitude 11 in the direction opposite to that of ⃗ PQ where P and Q are (1,3,2) and
Q (-1,0,6)
8. If a⃗ =2 i−^ ^j+2 k^ , ⃗b=2 i−
^ ^j+ k^ ⃗c =i−2^ ^j+ k^ find a unit vector∥¿ 3 a⃗ −2 ⃗b+ c⃗
9. Find the position vector of the midpoint of the vector joining points P(2, 3, 4) and Q (4,-1,2)
10. Write down two different vectors of same magnitude.
^
11. If ⃗a =i−2 ^j and b=3⃗ ^ 2 ^j−k^ , ⃗c =−4 i+6
i+ ^ ^j−8 k^ find |3 a⃗ −2 ⃗b+ 4 ⃗c|
12. Find the magnitude of the vector a⃗ =i+3 ^ ^j−5 k^
13. Find the values of x and y so that the vectors x i−2 ^ ^j+ k^ , 4 i+
^ y ^j+ k^ are equal.
14. Find the direction cosines of the vector a⃗ =−7 i−2 ^ ^j−2 k^
15. Write two different vectors with same direction.
16. Find the scalar and vector components of a vector whose initial point is (-1, -2,3) and terminal point is
(0, -5,6)
17. Show that the vectors a⃗ =2 i−3 ^ ^j+4 k∧
^ ⃗b=−4 i+6 ^ ^j−8 k^ are collinear.
18. If ⃗PO +⃗ OQ=⃗ QO+ ⃗¿ , then show that P, Q, R are collinear.
⃗ c be the vectors represented by the sides of triangle taken in order, then prove that a⃗ + ⃗b+ c⃗ =O
19. If a⃗ , b∧⃗ ⃗
20. Find the direction ratios of the vector a⃗ =4 i+7 ^ ^j+ k^ and hence find the direction cosines of the vector.
21. Find the angles at which the vector a⃗ =2 i−6 ^ ^j+ k^ is inclined to each of the co – ordinate axes.
22. Find the direction cosines of the vector joining points A (1,2, -3) and B (-1,-2,1) directed from A to B.
23. Show that the vector i+ ^ ^j+ k^ is equally inclined to the axes OX, OY and OZ.
24. If a vector makes angle x, y and z with the axes, show that sin2 x+ sin2 y+ sin2 z =1
25. The two vectors i+ ^ k^ ∧3 i−
^ ^j+4 k^ epresent the sides ⃗ AB∧⃗ AC . Find the unit vector parallel to ⃗
AB∧⃗ AC
26. ABCD is a parallelogram and if the co – ordinates are (1,2), (1,4), (0, -2) Find the fourth vertex of the
parallelogram.
27. If A (0,1), B (1,0), C (1,2) and D (2,1) then prove that ⃗ AB=⃗ CD
^ ^ ^ ^ ^
28. Show that the points 2i , −i−4 j ,−i+ 4 j form an isosceles triangle.
29. Find the position vector of the midpoint of the vector joining points P (2, -3,4) and Q (4,1, -2)
30. Find the vector from the origin to the centroid of the triangle whose vertices are (1, -1,2), (2,1,3),
(-1,2, -1)
31. If u⃗ =2 i−^ ^j+2 k^ , ⃗v =−2 i− ^ ^j−3 k^ ⃗w =4 i−3^ ^j+ k^ , x and y such that ⃗ w =x ⃗u + y ⃗v
32. using vectors, prove that the points (-2,1), (-5, -1) and (1,3) are collinear.
33. If the vectors a⃗ =2 i−3 ^ ^j , ⃗b=−6 i+m
^ ^j are collinear , find the value of m .
34. Find the components along the axes of the point (-3,5)
35. If a⃗ is the position vector whose tip is (5, -3), find the co – ordinates of B such that ⃗ AB=⃗a,
the co-ordinates of A being (4, -1)
36. Find the projection of 2 i− ^ ^j+2 k^ on−2 i− ^ ^j−3 k^
37. Find the projection of i+3 ^ ^j+¿ 38. Find |⃗x|, for a unit vector ⃗a ⃗ ( ⃗x −⃗a ) . ( ⃗x + ⃗a )=12
39. Find |⃗a|∧|⃗b|if ( ⃗a + ⃗b ) . ( a⃗ −b⃗ )=8
40. Find the angle between the vectors i−2 ^ ^j+3 k^ ∧3 i−2^ ^j+ k^
41. Evaluate: ( 3 a⃗ −4 ⃗b ) . ( 2 ⃗a +5 ⃗b )
^ ^j−3 k^ , ⃗b=i+3
42. a⃗ =5 i− ^ ^j−5 k^ ,then show that a⃗ + ⃗b∧⃗a− ⃗b are perpendicular.
43. State and prove Cauchy – Schwartz inequality.
44. State and prove triangle inequality,
45. Find |⃗a−b⃗| if 2 vectors are such that |⃗a|=2 ,|b⃗|=3 , ⃗a . ⃗b=4
46. Find |⃗a × ⃗b|if ⃗a=2 i− ^ ^j+2 k^ , ⃗b=−2 i− ^ ^j−3 k^
47. Find the area of parallelogram whose adjacent sides are a⃗ =3 i− ^ 2^j+2 k^ , ⃗b=2 i−^ ^j−3 k^
48. Given that a⃗ . ⃗b=12 ,|⃗a|=2 ,|⃗b|=3 , find|⃗a × ⃗b|
49.If a⃗ × ⃗b=⃗c × ⃗d∧⃗a × ⃗c =⃗b × ⃗d show that a⃗ −d⃗ is∥¿ b−⃗ ⃗ c
50. For any three vectors a⃗ , b⃗ , c⃗ are any three vectors , prove that a⃗ × ( ⃗b+ c⃗ ) + ⃗b × ( c⃗ +⃗a )+ ⃗c × ( ⃗ a+ ⃗b ) =⃗0
51. Write the value of ( i^ × ^j ) . k^ + ( ^j× k^ ) . i^
1 1 4
52. If |a⃗ × b⃗|= |⃗a|= ,|b⃗|= , find|⃗a . b⃗|
√3 2 √3
1 1 4
53. Find the angle between |⃗a × b⃗|= |⃗a|= ,|b⃗|= , find |⃗a . b⃗|
√3 2 √3
54. The two vectors are i+ ^ ^j∧3 i− ^ ^j+4 k^ represnt thetwo sides ⃗ AB∧⃗ AC Find the length of
the median through A.
55. Find the distance between the points P(2,3,-1) and Q ( 4,-3,0) using vector method.
56. If the position vector of a point (12, n) is such that |⃗a|=¿ 13, find n.
57. Find μ so that the vectors 2 i+ ^ μ ^j+ k^ ∧i−2 ^ ^j+3 k^ are perpendicular, find μ
58. Find the value of p so that the vectors are 3^i+ 2 ^j+ 9 k∧ ^ i−2
^ ^j+ 3 k^ are parallel and
perpendicular.
59. For any vector r⃗ =( ⃗r . i^ ) i+
^ ( ⃗r . ^j ) ^j+ ( r⃗ . k^ ) k
60. Let a⃗ ∧⃗b be two vectors such that they have same magnitude and angle between them is 600
and a⃗ . ⃗b=8, find |⃗a|,|⃗b|
61. Dot product of vectors with vectors 3^i+ 5 ^j , 2^i+ 7 ^j∧i+ ^ ^j+ k^ are -1, 6,5 . Find the vector.
62. If 2 vectors are such that |⃗a|=2 ,|b⃗|=2 , ⃗a . ⃗b=6 , find|⃗a + b⃗|∧|⃗a− ⃗b|

63. I f a^ ∧b^ are unit vectors inclined at an angle θ , then prove that sin θ =
|^a−b^|
2 2
64. If a^ ∧b^ are unit vectors inclined at an angle θ , then prove that cos θ =
|a^ + b^|
2 2
θ |a^ −b^|
65. If a^ ∧b^ are unit vectors inclined at an angle θ , then prove thattan =
2 |a^ + b^|
b , c⃗ are three mutually perpendicular vectors of magnitude3 , 4∧5 find|2 a⃗ + ⃗b+ c⃗|
66. If a⃗ , ⃗
b , c⃗ are three mutually perpendicular unit vectors prove that|⃗a + b+
67. If a⃗ , ⃗ ⃗ ⃗c|=√ 3
68. Find the angle between the vectors a⃗ ∧⃗b so that √ 3 a⃗ −b⃗ is a unit vector .
69. If a⃗ ∧⃗b are two vectors such that |⃗a + ⃗b|=|a⃗| prove that 2a⃗ + ⃗b is perpendicular ¿ b⃗
70. Find the values of x for which the angle between a⃗ =2 x i+4 ^ x ^j+ k^ , b=7
⃗ ^
i−2 ^j+ x k^ is obtuse.
71. Show that ( a⃗ −⃗ b ¿ ¿ ×(⃗a + ⃗b)= 2 (a⃗ × ⃗b)
72. Find α ∧β such that ( 2 i+ ^ 6 ^j+27 k^ ) ×( i+α ^ ^j+ β k^ ) = o⃗
2 2 2
73. For any vector a⃗ , prove that|⃗a × i^ | +|a⃗ × ^j| +|⃗a × k^| =2|⃗a|2
74. For any vector r⃗ =x i^ + y ^j + z ^k , prove that ( ⃗r × i^ ) . ( r⃗ × ^j ) + xy=0
75. Find the angle between a⃗ ∧⃗b , if |⃗a × ⃗b|=⃗a . ⃗b and also prove that |⃗a × ⃗b|=⃗a . ⃗b(tanθ)
THREE MARKS TYPE
^
1. One side of the parallelogram is given by 2i−4 ^j+5 k^ ∧the diagonal vector is3 i−6
^ ^j+2 k^ . Find
the unit vector parallel to the diagonal and also find the other side of the parallelogram. Find the area of
parallelogram.
2. Write all the unit vectors in xy plane. Obtain a unit vector in xy plane making an angle of 30 with
positive
direction of x axis.
3. Consider 2 points P and Q with PV ⃗ OP=2 ⃗a + ⃗b∧⃗ OQ=⃗a−3 ⃗b in which R divides PQ in the ratio
2 : 1 internally and externally. Find the position vector of R in both these cases. Find the midpoint of RQ
and show that P is the midpoint of RQ.
4. Show that the points A (1, -2,8), B (5,0,2) and C (11,3,7) are collinear. Find the ratio in which B divides
AC.
5. A vector r⃗ is inclined at equal acute angles to x, y, and z axes. If |r⃗|=6 units , find r⃗
6. A) Can a vector have direction angles 450, 600, 1200?
B) A vector r⃗ is inclined at 450 to x axis and 600 to y axis and |r⃗|=8 units , find r⃗
7. If a⃗ ∧⃗b determine two adjacent sides of a regular hexagon, what are the vectors determined by other
sides taken in order?
8. The PV of A, B, C are α i+3 ^ ^j , 12 i+ ^ μ ^j∧11 i−3 ^ ^j∧C∣AB∈the ratio 3 :1
Find α ∧μ
9. Show that the points A( -2,3,5), B(7,0,-1), C(-3,-2,-5) and D(3,4,7) are such that AB and CD intersects at
the point P(1,2,3)
10. Show that the points with position vectors a⃗ −2 ⃗b+3 c⃗ ,−2 ⃗a +3 b−⃗ ⃗ c , 4 ⃗a−7 b+
⃗ 7 ⃗c are collinear.
^ ^j ,12 i−5 ^ ^j∧a i+
^ 11 ^j are collinear , find a
Find ⋋.
11. If the points with PV 10i+3
12. If a⃗ =2 I^ +2 J^ +3 ^ K , ⃗b=− I^ +2 J^ + ^ K , ⃗c =3 I^ + J^ are such that a⃗ + ⋋ b⃗ is perpendicular to c⃗ ,
13. The scalar product of the vector ^I + J^ + ^ K with a unit vector along the sum of vectors 2 ^I + 4 J^ −5 ^
⋋ ^I + 2 J^ +3 K ^ is equal to one. Find ⋋.
K and

14. Prove that (a⃗ + ⃗b) (a⃗ + ⃗b) = │ ⃗a │2 +│ ⃗b │2 if and only if a⃗ , b⃗ are perpendicular and a⃗ ⧧ 0⃗ , ⃗b ⧧ ⃗0
15. Three Vector a⃗ , b⃗ and c⃗ satisfy the condition a⃗ + ⃗b+ c⃗ = ⃗0 . μ|=⃗a . ⃗b+ b⃗ . c⃗ + c⃗ . ⃗a if
│a⃗ │=3 , │ ⃗b│= 4, │c⃗ │=2.
16. Let a⃗ , b⃗ , c⃗ be three vectors such that │ ⃗a │=3 ,│ ⃗b │=4 , │ c⃗ │=5 and each of them being perpendicular
to other two, find │a⃗ + b+ ⃗ ⃗c │
17. If ^I + J^ + ^ K ,3 ^I +5 J^ ,3 ^I + 2 J^ −3 K^ , ^I −6 J^ − ^ K are P.V. of A, B, C and D. Find the angle between
⃗ ⃗
AB∧CD are collinear.
18. If the vertices of triangle ABC are (1,2,3), (-1,0,0), (0,1,2) Find ∟ABC.
19. If the sum of 2-unit vector is a unit vector, prove that the magnitude of their difference is √3.
20. If a⃗ , b⃗ , c⃗ are three mutually perpendicular vectors of magnitudes 3, 4, 5 find │2a⃗ + ⃗b+ c⃗ │
21. Leta⃗ , b⃗ , c⃗ be three vectors such that │ a⃗ │=1, │ ⃗b │=2, │ ⃗c │=3. If the projection of
b⃗ along ⃗a is equal ¿ theprojection of c⃗ along ⃗a and b⃗ , c⃗ is perpendicular to each other.
Find │3a⃗ −2 ⃗b+2 ⃗c 22.Find the angle between the vector a⃗ ∧⃗b so that √3a⃗ −⃗b is also a unit vector.
23.Ifa⃗ , b⃗ , c⃗ are three vectors such that ⃗a . b=⃗ ⃗ a . c⃗ , thenshow that ⃗a=0⃗ ∨⃗a perpendicular ¿ ¿
24. Show that the angle between the two diagonals of a cube is Cos-1(1/3).
25. Find the values of x for which the angle between a⃗ =2 x2 ^I + 4 x J^ + ^ ⃗
K and b=7 ^I −2 J^ + x ^
K is obtuse.
26. Find a unit vector perpendicular to each of a⃗ + ⃗b and a⃗ −⃗b where a⃗ = ^I + J^ + ^ ⃗
K , b=2 ^ ^j+ k^
i−
27. Find the area of triangle whose vertices are A (1,1,2), B(2,3,5) and C(1,5,5)
28. Let a⃗ =i+4 ^ ^j+2 k^ , b=3
⃗ ^
i−2 ^j+7 k^ , ⃗c =2 i−
^ ^j+ 4 k^ , find a vector d⃗ which is perpendicular to
both a⃗ ∧⃗b∧⃗c . d⃗ =15
29. For any vector a⃗ , prove that|⃗a × i^ |2 + |⃗a × ^j|2 + |⃗a × k^|2 = 2|⃗a|2
1⃗ ⃗
30. Prove that the area of quadrilateral ABCD is | AC × BD| where AC and BD are its diagonals.
2
31. If a⃗ , b⃗ , c⃗ are the position vectors of vertices A, B and C of triangle ABC, show that area of
1
∆ ABC= |⃗a × b⃗ + b⃗ × c⃗ + c⃗ × a⃗| . Deduce the condition for the points to be collinear.
2
32. Ifa⃗ , b⃗ , c⃗ are three vectors such that Ifa⃗ + ⃗b+ c⃗ =O ⃗ are three vectors , then prove that
Ifa⃗ × ⃗b= ⃗b × c⃗ =⃗c × ⃗a
33. If a⃗ =i+ ^ ^j+ k^ , find a vector b⃗ , satisfying the relation ⃗a × b=⃗
⃗ c ∧⃗a . ⃗b=3

34. Let a⃗ , b⃗ , c⃗ be unit vectors such that ⃗a . ⃗b=⃗a . ⃗c =0and angle between b∧⃗ ⃗ c is π
6
Then prove that a⃗ =± 2 ( ⃗b × c⃗ )
35. Ifa⃗ + ⃗b+ c⃗ =O ⃗ , │ ⃗a │ =3, │ ⃗b │=5, │ ⃗c │=7, then show that angle between a⃗ ∧⃗b is 600

FIVE MARKS TYPE

1. If a⃗ , b⃗ , c⃗ are mutually perpendicular vectors of equal magnitude , show that the vector ⃗a + b⃗ +⃗c is equally
inclined to a⃗ , b⃗ , c⃗
2. If a⃗ , b⃗ , c⃗ are vectors such that ⃗a . ⃗b=⃗a . ⃗c ⃗a × ⃗b=⃗a . ×⃗ c , ⃗a ≠ 0⃗ show that ⃗b=⃗c
3. If with reference to right-handed system of mutually perpendicular unit vectors, i^ , ^j, k^ where α⃗ =3 i−¿ ^
^j , ⃗β=2 i+^ ^j−3 k^ , then express ⃗β ∈the form ⃗ ⃗β=⃗ ⃗β 1+ β⃗ 2 where ⃗ ∝ and ⃗
β 1 is∥¿ ⃗ β2
is perpendicular to ⃗ ∝
π ^
i∧π
4. If a unit vector makes an acute angle with with ^j ∧an angleθ´ with k^ . Find θ
3 4
And components of a⃗
5. A vector r⃗ isinclined at equal angles ¿ three axes . If the magnitude of r⃗ is 2 √ 3 , find ⃗r
6. Show that the points A(3,-4,-4), B(2,-1,1) and C ( 1,-3,-5) form the vertices of right angled triangle using
vectors.
7. Find the value of a + b if (2,a,3), (3,-5,b) and (-1,11,9) are collinear.
8. ⃗a ∧⃗b are unit vectors such that ⃗a +3 b⃗ is perpendicular ¿ 7 ⃗a−5 b∧⃗ ⃗ a −4 ⃗b is perpendicular ¿ 7 ⃗a−2 ⃗b Find
angle between a⃗ ∧⃗b
9. If a⃗ =i+ ^ ^j+ k^, c⃗ = ^j−k^ are given vectors , find a vector b⃗ satisfying ⃗a . b=3∧⃗ ⃗ ⃗ c
a × b=⃗
10. Find the values of c for which a⃗ =( c log 2 x ) i−6 ^ ^j +3 k∧^ ⃗b=( log x ) i^ +2 ^j + ( c log x ) k^
2 2

makes an angle obtuse for x ∈(0 , ∞)

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