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Byju's Answer
Standard XII
Physics
Examples
Prove that fo...
Question
Prove that for any three vectors a, b, c [b + c c + a a + b] = 2 [abc].
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Solution
For any vector
∴
→
a
,
→
b
and
→
c
, prove that
[
→
a
+
→
b
→
b
+
→
c
→
c
+
→
a
]
=
2
[
→
a
→
b
→
c
]
sowing LHS :-
[
→
a
+
→
b
→
b
+
→
c
→
c
+
→
a
]
=
(
→
a
+
→
b
)
.
[
(
→
b
+
→
c
)
×
(
→
c
+
→
a
)
]
[
→
a
→
b
→
c
=
→
a
.
(
→
b
×
→
c
]
=
(
→
a
+
→
b
)
.
[
(
→
b
×
→
c
)
+
(
→
b
×
→
a
)
+
(
→
c
×
→
c
)
+
(
→
c
×
→
a
)
]
∵
[
→
c
×
→
c
=
|
→
c
|
|
→
c
|
sin
O
^
n
=
o
=
(
→
a
+
→
b
)
.
[
(
→
b
×
→
c
)
+
(
→
b
×
→
a
)
+
o
+
(
→
c
×
→
a
)
]
=
→
a
.
(
→
b
×
→
c
)
+
→
b
.
(
→
b
×
→
c
)
+
→
a
.
(
→
b
×
→
a
)
+
→
b
.
(
→
b
×
→
a
)
+
→
a
.
(
→
c
×
→
a
)
+
→
b
.
(
→
c
×
→
a
)
=
[
→
a
,
→
b
,
→
c
]
+
[
→
b
→
b
→
c
]
+
[
→
a
→
b
→
a
]
+
[
→
a
→
c
→
a
]
+
[
→
b
→
c
→
a
]
∵
[
→
b
→
b
→
c
]
=
[
→
c
→
b
→
b
]
=
→
c
.
(
→
b
×
→
b
)
As
(
→
b
×
→
b
)
=
→
o
=
→
c
.
o
=
→
o
∵
Using property
[
→
b
→
b
→
c
]
=
0
[
→
a
→
b
→
a
]
=
0
[
→
b
→
b
→
a
]
=
0
[
→
a
→
c
→
a
]
=
0
=
[
→
a
→
b
→
c
]
+
o
+
o
+
o
+
o
+
[
→
b
→
c
→
a
]
=
[
→
a
→
b
→
c
]
+
[
→
b
→
c
→
a
]
=
[
→
a
→
b
→
c
]
+
[
→
a
→
b
→
c
]
=
2
[
→
a
→
b
→
c
]
=
R
.
H
.
S
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Similar questions
Q.
Prove that for any three vectors a, b, c
[
b
+
c
c
+
a
a
+
b
]
=
2
[
a
b
c
]
.
Q.
Prove that for any three vectors
→
a
,
→
b
and
→
c
,
[
(
→
a
+
→
b
)
(
→
b
+
→
c
)
(
→
c
+
→
a
)
]
=
2
[
→
a
→
b
→
c
]
.
Q.
For any three vectors
→
a
,
→
b
and
→
c
, prove that vectors
→
a
−
→
b
,
→
b
−
→
c
and
→
c
−
→
a
are coplanar.
Q.
For any three vectors
→
a
,
→
b
and
→
c
, prove that
[
→
a
+
→
b
,
→
b
+
→
c
,
→
c
+
→
a
]
=
2
[
→
a
→
b
→
c
]
. Hence prove that the vectors
→
a
+
→
b
,
→
b
+
→
c
,
→
c
+
→
a
are coplanar. If and only if
→
a
,
→
b
,
→
c
are coplanar.
Q.
For any three vectors
¯
¯
¯
a
,
¯
¯
b
,
¯
¯
c
prove
[
¯
¯
¯
a
−
¯
¯
b
¯
¯
b
−
¯
¯
c
¯
¯
c
−
¯
¯
¯
a
]
=
0
.
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