Question

Prove that :-
ax(b+c)+bx(c+ā)+cx(ā+b)= 0
​

Answer 1

[axb bxc cxa] is nothing but a box of [a b c]

[axb bxc cxa] = (axb).((bxc)x(cxa))

which is a quadrapule product of four vetors

we have a formula for quadrapule product that is

(axb)x(cxd) = [a b d]c - [a b c]d

therefore applying this formula

(axb).((bxc)x(cxa)) = (axb).([b c a]c - [b c c]a)

[b c c] = 0 because when 2 vectors are equal out of three then the box of those 3 vectors will be 0

therefore

=(axb).([b c a]c) = ((axb).c)[b c a]

= [a b c][b c a]

box product follows commutative rule therefore [b c a] = [a b c] = [c a b]

=[a b c][a b c]

=[a b c]^2

hence proved :)