Question 24 (OR 1st Question) For three non-zero vectors a , b and c , prove that [ (a - b) (b - c) (c - a)] = 0
[ (a - b) (b - c) (c - a)]
= (a - b) (b - c) x (c - a)
= (a - b) [ (b x c) - (b x a) - (c x c) + (c x a)
(c x c) = [c] [c] sin 0° n
= 0
= (a - b). [(b x c) - (b x a) - 0 + (c x a)]
= (a - b). [ (b x c) - (b x a) - 0 + (c x a)]
= a [ (b x c) - (b x a) + (c x a) -b [(b x c) - (b x a) + (c x a)
= a . (b x c) -a . (b x a) + a (c x a) - b (b x c) b (b x a) - b (c x a)
= [a, b, c] - [a, b, a] + [a, c, a] - [b, b, c] + [b, b, a] - [ b, c, a]
now
[ b, b, c] = [c, b, b]
= c (b x b)
as (b x b) = 0
= c .0
= 0
similarly,
[a, b, a] = 0
[b, b, a] = 0
[a,c, a] = 0
= [a, b, c] - 0 + 0 - 0 + 0 - [ b, c, a]
= [a, b, c] - [ b, c, a]
as [a, b, c] = [c, a, b ] = - [b, c, a]
= [a, b, c] - [a, b, c]
= 0

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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