VECTORS

1. If ABCDEF is a regular hexagon and if and then
(1) (2)
(3) (4) 2

Solution: Ans(1).

2.The points with the position vectorsai 52j, 60i + 3jand 40i – 8j are collinear. The value of ‘a’ is
(1) 40 (2) 40 (3) 20 (4) 20Solution: Ans(2).
5
60 100 40.

3. If is a unit vector perpendicular to the unit vectors and and if the angle between and is then
(1) 0.50(2) 0.30(3) 0.25 (4) 0.40

Solution:Ans(3).
cos0sin 1.1. . ( Note: Here is parallel to )

4. The angle made by the vector (2i – 2j + k) with the positive direction of the y-axis is
(1) sin- 1(2) cos- 1
(3) cos- 1 (4) cos- 1 Solution:Ans(4). Since the magnitude of 2i – 2j + k is 3, the given vector is a unit vector. The direction cosines of the given vector are and . The angle made by the given vector with the positive direction of the y-axis is cos- 1.

5. The area of the triangle, two whose sides are the vectors i + j and i j inclined at an angle 30 is
(1) 0.50(2) 0.30(3) 0.25 (4) 0.40

Solution:Ans(1)Area of the triangle
sin30
.

6. If C is the midpoint of the line joining the points A and B and if P is any other point non collinear with A and B then always
(1)
(2) 2
(3)
(4)

Solution:Ans(2). By the midpoint formula . 2.

7. Direction cosines of a vector are , and and magnitude of the vector is 6 units. The vector is
(1) ( 1, 2, 3) (2) ( 1, 2, 1)
(3) ( 2, 4, 4) (4) (2, 4, 4 )

Solution:Ans(3)Required vector is
6( 2, 4, 4)

8. If ai + j + k, i + bj + k and i + j + ck are coplanar then a + b + c
(1) abc (2) abc 2 (3) 0 (4) abc + 2 Solution:Ans(4). 0
a( bc 1) 1( c 1 ) 1( 1 b ) 0
abc a c 1 1 b 0
a + b + c abc + 2.

9. If and are unit vectors and if then the angle between and is
(1) 60 (2) 30(3) 45 (4) 90

Solution:Ans(1).


60.

10. The unit vector perpendicular to i + j and j + k forming a right handed system is
(1) (2)
(3) (4) Solution:Ans(2).Cross product of the given two vectors is . Required unit vector is .

11. If 2i + j – 2k and i – 2j + 2k then the angle between and is
(1) 60 (2) 30 (3) 90 (4) 45

Solution:Ans(3). Here 3. Since,vectors and are orthogonal. Hence angle between and is 90.

12.If , and are unit vectors such that + + then . + . + .
(1) 3 (2) 1 (3) (4) Solution:Ans(4).
2(. + . + . ) 0 1 + 1 + 1 + 2(. + . + . )
. + . + . .

13.A unit vector perpendicular to i and coplanar with j and i + j is
(1) j (2) i (3) (4)

Solution:Ans(3). Now
i ( j ( i + j ) i ( ( j i) + ( j j ))
i ( k + ) j. This is the required unit vector. (concept : )

14. If (1 p)i + 2(1 + p )j + (3 + p)k and 3i +j are at right angles to each other then p
(1) 2.5 (2) 5 (3) 3 (4) 4

Solution:Ans(2). Dot product
(1 p).3 +2(1+ p).1+ ( 3 + p).0 0
5 p 0 p 5.

15. If then the angle between and is
(1) 30 (2) 60 (3) 45 (4) 90

Solution:Ans(3).cos and sin.
cos sin
45.

16.If i j and j k then unit vector perpendicular to the plane of ABC is
(1) (2)
(3) (4) Solution:Ans(4). Required is .

.
Required is

17. If A, B, C and D are the vertices of a parallelogram, P is the point of intersection of the two diagonals and O is the origin then
(1) 4 (2) 2 (3) (4)

Solution:Ans(1).By themidpoint formula, 2 and 2 4.

18.If and are the three coterminous edges of a parallelepiped having volume 3cubic units then
(1) 6 (2) 9 (3) 3 (4) 27

Solution:Ans(2).
32 9.

19.If , 0 then for a constant ,
(1) (2)
(3) (4)

Solution:Ans(3).


and are parallel
.

20.If and are two unit vectors inclined at an angle to each other then will be less than 1if is greater than
(1) 30 (2)75 (3)135 (4)120

Solution:Ans(4). 2cos
1 + 1 + 2cos 2 + 2cos
2( 1 + cos) 4cos2.
2cos.
This is less than one whencos or 60 or 120.

21. For what value of the vector 3i j + 5k lie on the plane determined by the vectors (2, 1, 1) and (1, 2, 3) ?
(1) 8 (2) 8 (3) 6 (4) 6

Solution:Ans(1). Three vectors are coplanar
0
3( 5 ) ( 5) 5( 5 ) 0
8.

22. The projection of 2i 3j on 3j 2k is
(1) (2) (3) (4)Solution:Ans(2).Required . 23. The three vectors 7i 11j k, 5i 3j 2k and 12i 8j k form the sides of
(1) an isosceles (2) an equilateral
(3) a right angled (4) a scalene

Solution:Ans(3). Magnitudes of the three vectors are , and . Since 171 + 38 209 the triangle formed by the vectors is right angled.

24.The unit vectors and are inclined at an angle 90. The magnitude of is
(1) 24 (2) 20 (3) 5 (4) 10

Solution:Ans(4).

9 10.

10sin90 10.

25.The points A, B and C having the position vectors i + 2j + 3k, i + 4j + 7k and 3i 2j 5k respectively are collinear iff
(1) 3 (2) 2 (3) 1 (4) 4

Solution:Ans(1). ( 1, 2, 4 ) and
( 4, 4, 8). Now and are parallel. 1 2. 3.

26. If is a non zero vector of magnitude 'a' units and is a scalar then m is a unit vector if a
(1) (2) (3) (4)

Solution:Ans(2). 1 1
a 1 a .

27. If and are the angles made by the vector 2i + 3j + 4k with the positive direction of the x-, y- and z-axes then sin2+ sin2+ sin2
(1) 1 (2) 4 (3) 2 (4)3

Solution:Ans(3). Formula: cos2+ cos2+ cos21. sin2+ sin2+ sin2
1 cos2+ 1 cos2+ 1 cos2
3 (cos2+ cos2+ cos2) 3 1 2.

28. The position vector of the centroid of the triangle formed by the points having position vectors i +2j + 3k, 2i – 5j + 3k and 3i + 6j – 12k is
(1) 3i + 2j – k (2)i – j + k
(3) 2i – j + 3k (4) 2i + j – 2k

Solution:Ans(4).The position vector of the centroid is
.

29. If A, B, C and D are the four points such that then
(1) 2 (2) 2 (3) 2 (4) 2

Solution:Ans(1). By the triangle law of addition of the vectors,
+
.

30.Which of the following is FALSE? (Here and are unit vectors with the usual meaning)
(1)(2)0
(3) 0(4)

Solution: Ans(2).
.
3.
0.

) . 31. If and and if 1 and then angle between and is
(1) cos– 1 (2) cos – 1
(3) cos – 1 (4) cos – 1 Solution:Ans(3). 0
0.
3 – 11 + 6 0
cos cos
cos – 1 . 32.If and are the position vectors of the four points A, B, C and D such that 0 and
0 then for the ABC the point D is
(1) centroid (2) incenter
(3)orthocenter (4) circumcenter

Solution:Ans(4). Let E and F be the midpoints of BC and AC. By the given condition 0 and 0. and This shows that D is the circumcenter of the ABC.

33. If and are three vectors such that each one is perpendicular to the sum of the other two and if 3, 4 and 5 then
(1) (2) (3) (4)

Solution:Ans(1). Given, and . 0, 0 and 0. 0 2( ) 0.

9 + 16 +25 50.
.

34. If p, q, r are direction cosines of a vector perpendicular to the vector 2i – 3j + 4k then
(1) p + q + r 0 (2) 2p – 3q + 4r 0
(3) 0 (4)

Solution: Ans(2). The unit vector pi + qj + rkis perpendicular to the vector 2i – 3j + 4k. the dot product of the vectors is zero.

35. If and are two vectors of magnitudes 2 units each then
(1) 0 (2) 4 (3) 16 (4) 8

Solution:Ans(3).
2( ) 2( 4 + 4) 16.

36. If the angle made by 2i + 2j – k with the positive direction of x-axis is 60 then
(1) (2) (3) (4)

Solution:Ans(4). sin60
3.1. .

37. If and are three mutually perpendicular vectors each of magnitudes 3 units the
(1) 27 (2) 18 (3) 9 (4) 6

Solution:Ans(1).
cos0sin90
3.3.3 27.( Here is parallel to )

38.The vertices A, B and D of a parallelogram ABCD are
i +2j, i + 2k and k + 2i. The point C is
(1)i – 3j + k (2) 2i – 2j + 2k (3) 3i – 2j + 3k (4)i – j + k

Solution:Ans(2). Let C (x, y, z ).Midpoint of AC is same as the midpoint of BD. i +2j + xi + yj + zk i + 2k + k + 2i
xi + yj + zk 2i – 2j + 3k

39.
(1)( ) (2)( ). (3)( ) (4)( ).

Solution:Ans(3).

required
() ()
( ) ( )
( ) ( )
( ).

40. If i 2j 3k then
(1) xi + j k (2) xi 3j 2k
(3) xi j k (4) xi + 3j 2k

Solution:Ans(4).Let xi + yj + zk. Then
i
.
y 3 and z 2. xi + 3j 2k.