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Byju's Answer
Standard XII
Mathematics
Collinear Vectors
Position vect...
Question
Position vectors of
A
,
B
,
C
are given by
→
a
,
→
b
,
→
c
and they are collinear, prove that
(
→
a
×
→
b
)
+
(
→
b
×
→
c
)
+
(
→
c
×
→
a
)
=
0
.
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Solution
If
A
,
B
,
C
are collinear
→
A
B
and
→
B
C
are parallel
⇒
(
→
b
−
→
a
)
×
(
→
c
−
→
b
)
=
0
⇒
(
→
b
−
→
a
)
×
→
c
−
(
→
b
−
→
a
)
×
→
b
=
0
⇒
(
→
b
×
→
c
)
−
(
→
a
×
→
c
)
−
(
→
b
×
→
b
)
+
(
→
a
×
→
b
)
=
0
⇒
(
→
b
−
→
a
)
×
→
c
−
(
→
b
−
→
a
)
×
→
b
=
0
⇒
(
→
b
×
→
c
)
−
(
→
a
×
→
c
)
−
(
→
b
×
→
b
)
+
(
→
a
×
→
b
)
=
0
But
→
b
×
→
b
=
0
∴
(
→
b
×
→
c
)
−
(
→
a
×
→
c
)
+
(
→
a
×
→
b
)
=
0
∴
(
→
b
×
→
c
)
+
(
→
c
×
→
a
)
+
(
→
a
×
→
b
)
=
0
Hence proved.
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Similar questions
Q.
Prove that points
A
,
B
,
C
having positions vectors
→
a
,
→
b
,
→
c
are collinear, if
[
→
b
×
→
c
+
→
c
×
→
a
+
→
a
×
→
b
]
=
→
0
Q.
If
→
A
,
→
B
&
→
C
are vectors such that
|
→
B
|
=
|
→
C
|
, Prove that :
[
(
→
A
+
→
B
)
×
(
→
A
+
→
C
)
]
×
(
→
B
×
→
C
)
.
(
→
B
+
→
C
)
=
0
Q.
If
→
A
,
→
B
and
→
C
are vectors such that
|
→
B
|
=
|
→
C
|
, prove that
[
(
→
A
+
→
B
)
×
(
→
A
+
→
C
)
]
×
(
→
B
×
→
C
)
.
(
→
B
+
→
C
)
=
0
.
Q.
If
→
a
,
→
b
,
→
c
are three vectors such that
→
a
×
→
b
=
→
c
,
→
b
×
→
c
=
→
a
,
→
c
×
→
a
=
→
b
then prove that
|
→
a
|
=
|
→
b
|
=
|
→
c
|
Q.
If
→
a
+
→
b
+
→
c
=
0
, the prove that:
→
a
×
→
b
=
→
b
×
→
c
=
→
c
×
→
a
where
→
a
,
→
b
,
→
c
are non-zero vectors.
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