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 ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image005.gif)
+
will be a unit vector
if and only if ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image006.gif)
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![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image012.gif)
&⇒ 2
= - 3
< 0
&⇒ 2 ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image014.gif)
&⇒ cosq
< 0
&⇒ q >
90°
i.e. ∠
BOA > 90°.
Question.3
If for non zero vectors
then
(A)
is
perpendicular to the plane of
and ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image016.gif)
(B)
is
parallel to the plane of
and ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image016.gif)
(C)
is
perpendicular to
and parallel to
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image016.gif)
(D) None of these
Solution
&⇒ ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image018.gif)
&⇒
is
collinear with ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image019.gif)
&⇒
is parallel to the
plane of ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image020.gif)
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) ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image023.gif)
(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
.
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image028.gif)
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) ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image032.gif)
(C)
(D) None of these
Solution
Let ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image034.gif)
&⇒ a2
+ b2 = 1 …(1)
Also,
cos45° = ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image035.gif)
&⇒ a
+ b = 1 …(2)
cos60° = ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image036.gif)
&⇒ 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
.
= ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image043.gif)
&⇒ 0
= n + 2![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image044.gif)
&⇒
= ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image045.gif)
Question.7
Find the angle
between the two straight lines
and
.
(A)
(B) ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image049.gif)
(C)
(D)
Solution
The first line
parallel to the vector ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image052.gif)
The second line
parallel to the vector ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image053.gif)
If q is the angle between the lines then,
= ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image055.gif)
\
.
Question.8
If
, then
´[
´
]is equal to
(A) A vector perpendicular to plane of ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image061.gif)
(B) A scalar quantity
(C) ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image062.gif)
(D) None of these
Solution
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image063.gif)
&⇒
Vectors
are coplanar
&⇒
and
are collinear
&⇒
.
Question.9
If r, a, b, c are
non null vectors such that
, then ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image069.gif)
(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
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image084.gif)
&⇒ ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image085.gif)
=
8
&⇒
= 8
&⇒ ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image088.gif)
&⇒ ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image089.gif)
&⇒
= 1. 3.
= ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%202_files/image091.gif)