Question

For 3 non zero vectors a b c prove that [a-b b-c c-a]=0

Answer 1

=[a-b, b-c, c-a] =0

=[a-b, b-c. c-a]

={(a-b) ×(b-c)}. ( c-a)

=(a×b-a×c+b×c. ( c-a)

=(a×b+c×a+b×c). (c-a)

=(a×b).c-(a×b).a+(c×a).c-(c×a).a+(b×c).c-(b×c).a

=[abc] - [aba] +[cac] - [caa] +[bcc] - [bca] +[abc] -

[bca]

=[abc] - [abc] =0

Answer 2

Step-by-step explanation:

=[a-b, b-c, c-a] =0

=[a-b, b-c. c-a]

={(a-b) ×(b-c)}. ( c-a)

=(a×b-a×c+b×c. ( c-a)

=(a×b+c×a+b×c). (c-a)

=(a×b).c-(a×b).a+(c×a).c-(c×a).a+(b×c).c-(b×c).a

=[abc] - [aba] +[cac] - [caa] +[bcc] - [bca] +[abc] -

[bca]

=[abc] - [abc] =0