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.
= 