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Solved Objective Question on Vector Set 2

Posted on - 04-01-2017

Math

IIT JEE

Question.1

Let and be two unit vectors such that + is also a unit vector . The angle between and is

(A) 30°

(B) 60°

(C) 90°

(D) 120°

Solution

Let q be the angle between and + will be a unit vector if and only if

i.e. .= – 1/2 or cosq = – 1/2 or q = 1200

Question.2

are two vectors such that =. Then;

(A) ∠BOA = 90°

(B) ∠BOA > 90°

(C) ∠BOA < 90°

(D) 60° £ ∠BOA £ 90°

Solution

Given =.
On squaring (OA)2 +(OB)2 +2= (OA)2 + 4(OB)2 +4


&⇒
2= - 3< 0
&⇒
2


&⇒
cosq < 0
&⇒
q > 90° i
.e. ∠ BOA > 90°.

Question.3

If for non zero vectors then

(A) is perpendicular to the plane of and

(B) is parallel to the plane of and

(C) is perpendicular to and parallel to

(D) None of these

Solution


&⇒


&⇒
is collinear with
&⇒
is parallel to the plane of

Question.4

If are two unit vectors and q is the angle between them, then the unit vector along the angular bisector of will be given by

(A)

(B)

(C)

(D) none of these. .

Solution

vector in the direction of angular bisector of = .

have magnitude cos(q/2)

so, the unit vector in this direction will have magnitude .

Question.5

A unit vector in xy – plane which makes an angle of 45° with the vector and an angle of 60° with the vector is

(A)

(B)

(C)

(D) None of these

Solution

Let
&⇒
a2 + b2 = 1 …(1)

Also,

cos45° =
&⇒
a + b = 1 …(2)

cos60° =
&⇒
3a – 4b = …(3)

There exists no real values of a and b satisfying (1), (2) and (3).

Hence no such unit vector exists.

Question.6

If where || = 1 " i, then the value of is

(A) –n/2

(B) –n

(C) n/2

(D) n

Solution

= 0

.=
&⇒
0 = n + 2


&⇒
=

Question.7

Find the angle between the two straight lines and .

(A)

(B)

(C)

(D)

Solution

The first line parallel to the vector

The second line parallel to the vector

If q is the angle between the lines then, =

\ .

Question.8

If , then ´[´]is equal to

(A) A vector perpendicular to plane of

(B) A scalar quantity

(C)

(D) None of these

Solution


&⇒
Vectors are coplanar


&⇒
and are collinear
&⇒
.

Question.9

If r, a, b, c are non null vectors such that , then

(A) is equal to 1

(B) cannot be evaluated

(C) is equal to zero

(D) none of these

Solution

Since = 0, and , must be perpendicular to all the three vectors . Hence must be coplanar


&⇒
=0

Question.10

Let and . If is a vector such that =and the angle between and is 300, then find the value of .

(A) 2/3

(B) 3/2

(C) 2

(D) 3

Solution

= sin 300


&⇒

= 8
&⇒
= 8


&⇒

&⇒

&⇒
= 1. 3. =

 
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