For any three vectors $¯ ¯ ¯ a, ¯ ¯ b, ¯ ¯ c$ prove
$[¯ ¯ ¯ a - ¯ ¯ b ¯ ¯ b - ¯ ¯ c ¯ ¯ c - ¯ ¯ ¯ a] = 0$ .

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Solution

$[\to a - \to b \to b - \to c \to c - \to a] = (\to a - \to b) \times (\to b - \to c) . (\to c - \to a)$
$= \to a \times \to b - \to a \times \to c - \to b \times \to b + \to b \times \to c) . (\to c - \to a)$
$= (\to a \times \to b - \to a \times \to c + \to b \times \to c) . (\to c - \to a) (∵ \to b \times \to b = 0)$
$= \to a \times \to b . \to c - \to a \times \to c . \to c + \to b \times \to c . \to c - \to a \times \to b . \to a + \to a \times \to c . \to a - \to b \to c . \to a$
$= \to a \times \to b \times \to - 0 + 0 - 0 + 0 - \to b \times \to c \times \to a$
$= [\to a \to b \to c] - [\to b \to c \to a]$
$= [\to a \to b \to c] - [\to a \to b \to c] = 0$

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