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Byju's Answer
Standard XII
Mathematics
Equality of Matrices
Show that a...
Question
Show that
→
a
,
→
b
,
→
c
are coplanar if and only if
(
→
a
+
→
b
)
,
(
→
b
+
→
c
)
and
(
→
c
+
→
a
)
are coplanar.
Open in App
Solution
→
a
,
→
b
,
→
c
a
r
e
c
o
p
l
a
n
a
r
.
⇔
[
→
a
→
b
→
c
]
=
0
⇔
2
[
→
a
→
b
→
c
]
=
0......
(
1
)
L
H
S
=
[
→
a
+
→
b
→
b
+
→
c
→
c
+
→
a
]
=
(
→
a
+
→
b
)
⋅
[
(
→
b
+
→
c
)
×
(
→
c
+
→
a
)
]
=
(
→
a
+
→
b
)
⋅
[
→
b
×
→
c
+
→
b
×
→
a
+
→
c
×
→
c
+
→
c
×
→
a
]
=
(
→
a
+
→
b
)
⋅
[
(
→
b
×
→
c
)
−
(
→
a
×
→
b
)
+
(
→
c
×
→
a
)
]
{
a
s
→
c
×
→
c
=
0
a
n
d
→
b
×
→
a
=
−
→
a
×
→
b
}
=
→
a
⋅
(
→
b
×
→
c
)
−
→
a
⋅
(
→
a
×
→
b
)
+
→
a
⋅
(
→
c
×
→
a
)
+
→
b
⋅
(
→
b
×
→
c
)
−
→
b
⋅
(
→
a
×
→
b
)
+
→
b
⋅
(
→
c
×
→
a
)
=
[
→
a
→
b
→
c
]
−
[
→
a
→
a
→
b
]
+
[
→
a
→
c
→
a
]
+
[
→
b
→
b
→
c
]
−
[
→
b
→
a
→
b
]
+
[
→
b
→
c
→
a
]
=
[
→
a
→
b
→
c
]
+
[
→
b
→
c
→
a
]
=
[
→
a
→
b
→
c
]
+
[
→
a
→
b
→
c
]
{
a
s
[
→
b
→
c
→
a
]
=
[
→
a
→
b
→
c
]
}
=
2
[
→
a
→
b
→
c
]
=
R
H
S
H
e
n
c
e
,
[
→
a
+
→
b
→
b
+
→
c
→
c
+
→
a
]
=
2
[
→
a
→
b
→
c
]
s
o
,
f
r
o
m
(
1
)
,
w
e
g
e
t
[
→
a
+
→
b
→
b
+
→
c
→
c
+
→
a
]
=
0
⇒
→
a
+
→
b
,
→
b
+
→
c
,
→
c
+
→
a
a
r
e
c
o
p
l
a
n
a
r
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0
Similar questions
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Show that vectors
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Show that the vectors
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