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Condition for Coplanarity of Four Points

Show that the...

Question

Show that the vectore $\to a$ , $\to b$ and $\to c$ are coplanar, if $\to a$ + $\to b$ , $\to b$ + $\to c$ and $\to c$ + $\to a$ are coplanar.

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Solution

Given
$\to a, \to b, \to c$ are coplanar
$\to a \cdot (\to b \times \to c) = 0$
$[\to a \to b \to c] = 0$ ----------(1)
To show
$(\to a + \to b) \cdot ((\to b + \to c) \times (\to c + \to a)) = 0$
$(\to a + \to b) \cdot (\to b \times \to c + \to b \times \to a + \to c \times \to c + \to c \times \to a) = 0$
$\to a \cdot (\to b \times \to c) + \to a \cdot (\to b \times \to a) + \to a \cdot (\to c \times \to c) + \to a \cdot (\to c \times \to a) + \to b \cdot (\to b \times \to c) + \to b \cdot (\to b \times \to a) + \to b \cdot (\to c \times \to c) + \to b \cdot (\to c \times \to a) = 0$
$[\to a \to b \to c] + [\to a \to b \to a] + [\to a \to c \to c] + [\to a \to c \to a] + [\to b \to b \to c] + [\to b \to b \to a] + [\to b \to c \to c] + [\to b \to c \to a] = 0$
but we know by definition of scalar triple product if two vectors is same in the triple scalar product then it is equal to zero
$[\to a \to a \to b] = 0$
SO
$[\to a \to b \to c] + 0 + 0 + 0 + 0 + 0 + 0 + [\to b \to c \to a] = 0$
$2 [\to a \to b \to c] = 0$
from eq (1)
$0 = 0$
Hence proved

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Similar questions

Q. Prove that the vector

\to a, \to b, \to c

are coplanar if

\to a + b, \to b + c, \to c + a

\to a, \to b, \to c

are non coplanar vectors such that

\to b \times \to c = \to a, \to c \times \to a = \to b

\to a \times \to b = \to c

\to a, \to b, \to c

are non-coplanar non-zero vectors such that

\to b \times \to c = \to a

\to a \times \to b = \to c

\to c \times \to a = \to b

\to a, \to b, \to c

are non-coplanar vectors, then

[\to a \times \to b

\to b \times \to c

\to c \times \to a] =

Q. For any three vectors

\to a, \to b

\to c

[\to a + \to b, \to b + \to c, \to c + \to a] = 2 [\to a \to b \to c]

. Hence prove that the vectors

\to a + \to b, \to b + \to c, \to c + \to a

are coplanar. If and only if

\to a, \to b, \to c

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Condition for Coplanarity of Four Points

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