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Question
Mathematics
If the vectors b , c , d are not coplanar, then the vector ( a × b ) ×( c × d )+( a × c ) ×( d × b ) +( a × d ) ×( b × c ) is
Q. If the vectors
b
,
c
,
d
are not coplanar, then the vector
(
a
×
b
)
×
(
c
×
d
)
+
(
a
×
c
)
×
(
d
×
b
)
+
(
a
×
d
)
×
(
b
×
c
)
is
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A
parallel to
a
B
parallel to
b
C
parallel to
c
D
perpendicular to
a
Solution:
The vectors
(
a
×
b
)
×
(
c
×
d
)
=
[
a
c
d
]
b
−
[
b
c
d
]
a
(
a
×
c
)
×
(
d
×
b
)
=
[
a
d
b
]
c
−
[
c
d
b
]
a
and
(
a
×
d
)
×
(
b
×
c
)
=
[
a
d
c
]
b
−
[
a
d
b
]
c
∴
(
a
×
b
)
×
(
c
×
d
)
+
(
a
×
c
)
×
(
d
×
b
)
+
(
a
×
d
)
×
(
b
×
c
)
=
[
a
c
d
]
b
−
[
b
c
d
]
a
+
[
a
d
b
]
c
−
[
c
d
b
]
a
+
[
a
d
c
]
b
−
[
a
d
b
]
c
[
∵
[
a
c
d
]
=
−
[
a
d
c
]]
=
−
2
[
b
c
d
]
a
∴
Given vector is parallel to
a
.