Prove that A-B-C are position vectors and a-b- and c- respectively are collinear if and only if a-b-b-c-c-a-0-

If A,B,C are collinear. A⃗B⃗ and B⃗C⃗ are parallel. ⇒A⃗B⃗×B⃗C⃗=0⃗ ⇒(b⃗ a⃗)×(c⃗ b⃗)=0⃗ ⇒(b⃗ a⃗)×c⃗ (b⃗ a⃗)×b⃗=0⃗ ⇒(b⃗×c⃗) (a⃗ ...

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Collinear Vectors

Prove that ...

Question

Prove that $A, B, C$ are position vectors and $\to a, \to b$ and $\to c$ respectively are collinear if and only if $(\to a \times \to b) + (\to b \times \to c) + (\to c \times \to a) = \to 0$

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A, B, C

\to a, \to b, \to c

[\to b \times \to c + \to c \times \to a + \to a \times \to b] = \to 0

A, B, C

\to a, \to b, \to c

(\to a \times \to b) + (\to b \times \to c) + (\to c \times \to a) = 0

a, b

c

\to a + \to b + \to c = \to 0

| \to a | = 7, | \to b | = 5

| \to c | = 3

\to b

\to c

\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]

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

\to a, \to b, \to c

\to a, \to b, \to c

\frac{| (\to a \times \to b) + (\to b \times \to c) + (\to c \times \to a) |}{| \to b - \to a |}

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