Question.1
then
(A)
(B) ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image003.gif)
(C)
(D) none of these
Solution
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image005.gif)
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image006.gif)
Question.2
The points with position vector 60i +
3j, 40i – 8j and ai –52j are collinear if
(A) a = -40
(B) a = 40
(C) a = 20
(D) none of these
Solution
The points are collinear
&⇒ l(
60i + 3j ) + m( 40 i- 8j) + n( ai- 52j) = 0 with l+m+
n = 0
&⇒ 60 l + 40 m
+ na = 0
3 l
- 8m - 52n = 0
l
+ m + n = 0
For non-zero set ( l, m,
n) ,
= 0
&⇒ a = -40
Question.3
are three vectors
respectively given by 2i +k, i + j+ k, and
4 i – 3j + 7k. Then vector
,
which satisfies the relation
, is
(A) 2 i – 5j + 2k
(B) –i + 4j + 2k
(C) -i
– 8j + 2k
(D) none of these
Solution
We have ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image011.gif)
&⇒
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image012.gif)
&⇒ ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image013.gif)
&⇒
( 2 +1)
= 3
– ( 8+7) ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image016.gif)
&⇒
=
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image017.gif)
Question.4
If the lines
intersect ( t and s are
scalars) then.
(A)
(B) ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image020.gif)
(C)
(D) none of these
Solution
For the point of intersection of the lines
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image022.gif)
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image023.gif)
Question.5
The number of vectors of unit length
perpendicular to vector a º
( 1, 1, 0) and b º ( 0, 1, 1) is
(A) 1
(B) 2
(C) 3
(D) 4
Solution
The vector of unit
length perpendicular to the given vectors
=
.
Hence there are two such vectors . .
Question.6
Let
º 2 i – j + k ,
º i+ 2j – k and
= i + j – 2k be three
vectors. A vector in the plane of
and
whose projection on
is of magnitude
is
(A) 2i + 3j – 3k
(B) 2i + 3j + 3k
(C) – 2i - j + 5k
(D) 2i + j + 5k
Solution
Let
be
a vector in the plane of b and c
&⇒
= ( i + 2j – k) + m ( i + j – 2k)
Its projection on
=
=
[2 +2 m - 2 - m -1 - 2m ] = ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image032.gif)
&⇒
= ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image033.gif)
&⇒ -( 1+m) = ±
2
&⇒ m = 1, -3
&⇒
R º 2 i + 3j – 3k and
- 2i – j + 5k .
Question.7
The scalar
equals
(A) 0
(B) ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image035.gif)
(C)![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image036.gif)
(D) none of these
Solution
= ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image039.gif)
&⇒
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image040.gif)
Question.8
Let
be
two unit vectors such that
is also
a unit vector. Then the angle between
is
(A) 30°
(B) 60°
(C) 90°
(D) 120°
Solution
Since
is
a unit vector, (
).(
) =1
&⇒
+
+ 2
=1
&⇒ 1 +1 + 2
=1
&⇒
= -1/2
Hence the angle q between
is
given by cosq = -1/2
&⇒
q =120°.
Question.9
If vectors ax
and
x
make an acute angle with
each other, for all x ∈ R, then a belongs
to the interval
(A)
(B) (
0, 1)
(C)
(D) ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image054.gif)
Solution
Since vectors make an acute angle with each
other so their dot product must be positive. .
i.e. ax2 – 10 ax + 6 > 0 " x ∈ R.
&⇒-ax2
+ 10ax – 6 < 0 " x ∈ R
&⇒-a
< 0 and 100a2 < 24a
Question.10
If
are unit vectors, such that
= 0, then the value of
is
(A) 1
(B) -1
(C) -3/2
(D) None of these
Solution
DABC
is an equilateral
\![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image058.gif)
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image059.gif)
Alternate
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image060.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image061.gif)
&⇒
=
– ![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image062.gif)
![](http://www.quizsolver.com/radix/dth/notif/VEC%20OBJ%20SOL%201_files/image063.gif)