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Byju's Answer
Standard IX
Mathematics
Unit Vectors
The two vecto...
Question
The two vectors
[
1
1
1
]
and
[
1
a
a
2
]
where a=
−
1
2
+
j
√
3
2
and j =
√
−
1
are
A
orthonormal
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B
orthogonal
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C
parallel
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D
collinear
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Solution
The correct option is
B
orthogonal
Let
X
1
=
⎡
⎢
⎣
1
1
1
⎤
⎥
⎦
a
n
d
X
2
=
⎡
⎢
⎣
1
a
a
2
⎤
⎥
⎦
=
⎡
⎢
⎣
1
w
w
2
⎤
⎥
⎦
∴
a
=
1
2
+
i
√
3
2
=
w
(Cube root of unity)
Now
X
1
.
X
2
=
1
×
1
+
1
×
w
+
1
×
w
2
=
1
+
w
+
w
2
=
0
Hence they are orthogonal.
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Similar questions
Q.
Two vectors
(
x
2
−
1
)
^
i
+
(
x
+
2
)
^
j
+
x
2
^
k
and
2
^
i
−
x
^
j
+
3
^
k
are orthogonal
Q.
→
A
=
2
^
i
+
3
^
j
and
→
B
=
3
^
i
+
2
^
j
are two vectors. Find the angle between the two vectors.
Q.
Two particles having position vectors
→
r
=
(
3
→
i
+
5
j
)
m
and
→
r
2
=
(
−
5
i
+
3
j
)
m
are moving with velocities
→
V
1
=
(
4
^
i
−
4
^
j
)
m
s
−
1
and
→
V
2
=
(
a
^
i
−
3
^
j
)
m
s
−
1
. If they collide after
2
seconds , the value of
a
is
Q.
If
¯
z
1
=
a
^
i
+
b
^
j
and
¯
z
2
=
c
^
i
+
d
^
j
are two vectors in
^
i
and
^
j
system where
|
¯
z
1
|
=
|
¯
z
2
|
=
r
and
¯
z
1
⋅
¯
z
2
=
0
, then
¯
ω
1
=
a
^
i
+
c
^
j
and
¯
ω
2
=
b
^
i
+
d
^
j
satisfy
Q.
The two vectors
(
x
2
−
1
)
^
i
+
(
x
+
2
)
^
j
+
x
2
^
k
&
2
^
i
−
x
^
j
+
3
^
k
are orthogonal.
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