For any three vectors a- b and c- prove that vectors a - b- b-c and c-a are coplanar-

If 3 vectors say A , B , C are coplaner. [ A B C #160;] =0 =[ a b b c c a #160;] =( a b #160;) .[ ( b ...

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Condition for Coplanarity of Four Points

For any three...

Question

For any three vectors $\to a, \to b$ and $\to c$ , prove that vectors $\to a - \to b, \to b - \to c$ and $\to c - \to a$ are coplanar.

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Solution

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\to a, \to b

\to c

\to p, \to q

\to r

\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}, \to q = \frac{\to c \times \to a}{[\to a \to b \to c]}

\to r = \frac{\to a \times \to b}{[\to a \to b \to c]}

(\to a + \to b) \cdot \to p + (\to b + \to c) \cdot \to q + (\to c + \to a) \cdot \to r =

\to a, \to b, \to c

\frac{a . (b \times c)}{(c \times a) . b} + \frac{b . (a \times c)}{c . (a \times b)} = ?

\to a, \to b, \to c

\to a + \to b + \to c

\to a, \to b

\to c

\to a, \to b, \to c

λ

[λ (\to a + \to b), λ^{2} \to b, λ \to c] = [\to a, \to b + \to c, \to b]

\to a, \to b, \to c

\to d

\to a + \to b + \to c

\to d = x \to b \times \to c + y \to c \times \to a + z \to a \times \to b

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