Prove without using components: If a *(b*c)=0 then (a*b)*(a*c)=0?

Question

Thanks!

Answer 1

If Ax(BxC)=0, then either A=0, B=0, or C=0, and therefore, any multiplication of combination of at least one of all of those three numbers would equal zero due to the multiplicative property of zero which states that any number or numbers, real or imaginary, multiplied by zero, equal zero, no matter what order the numbers are multiplied.

Answer 2

If these are just all scalars (numbers) a*(b*c) = 0 means that either a = 0, b = 0, and/or c = 0. This would imply that (a*b) and/or (a*c) = 0 meaning that the second expression is zero as well.

If these are vector dot products (b*c) would give you a scalar which must be zero for the expression a* (b*c) to equal zero. The second expression does not have to be zero and is only zero if the scalars (a*b) and/or (a*c) are zero.

Answer 3

Any number x 0 = 0. Either a, b, or c has to be 0 in the first statement... so the answer to the second statement has to be 0.

Answer 4

Simply because for the first equation to be true, one or more of a, b and c must be zero. If you are then using the same terms in the second equation, it too must be zero.

Answer 5

try to get a*(b*c) in the second expression, use the fact that multiplication is associative

Answer 6

My friend, you are absolutely correct, now the hard part, prove me wrong......