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Manipal Entrance Test 2015 Math Paper
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© examsnet.com
Question : 49
Total: 70
Let a, b and c be non-zero vectors such that no two are collinear and
(
a
×
b
)
×
c
=
1
3
|
b
|
|
c
|
a
.
If
θ
is the acute angle between the vectors b and
c
, then sin
θ
equals
2
√
2
3
√
2
3
2
3
1
3
Validate
Solution:
We have,
(
a
×
b
)
×
c
=
1
3
|
b
|
|
c
|
a
⇒
(
a
.
c
)
b
−
(
b
.
c
)
a
=
1
3
|
b
|
|
c
|
a
⇒
(
a
.
c
)
b
−
{
(
b
.
c
)
+
1
3
|
b
|
|
c
|
}
a
=
0
⇒
a
.
c
=
0
and
b
.
c
+
1
3
|
b
|
|
c
|
=
0
[∵ a, b are non-collinear]
⇒
|
b
|
|
c
|
cos
θ
+
1
3
|
b
|
|
c
|
=
0
⇒
cos
θ
=
−
1
3
⇒
sin
θ
=
√
1
−
cos
2
θ
=
√
1
−
(
−
1
3
)
2
=
√
1
−
1
9
=
√
8
9
=
2
√
2
3
© examsnet.com
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