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  4. If a, b and c are three non-coplanar vectors, then (a + b - c) · [(a - b) × (b - c) equals

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Q. If a, b and c are three non-coplanar vectors, then $(a + b - c) · [(a - b) × (b - c)$ equals

VITEEEVITEEE 2015

Solution:

$[a + b - c] . [(a - b) × (b - c)$
$= (a + b - c). [a × b - a × c - b × b + b × c]$
$= a. (a × b) - a. (a × c) + a. (b × c) + b.$
$(a × b) - b (a × c) + b. (b × c) - c. (a × b)$
$+ c. (a × c) - c.(b × c)$
$= a. (b × c) - b. (a × c) - c. (a × b)$
$= [a\, b \,c] - [b\, a \,c] - [c \,a \,b]$
$=[a \,b \,c] + [a\, b\, c] - [a\, b\, c]$
$= [a\, b \,c] = a. (b × c)$