Lin Alg Question: Cross-Product Proof

[ChaoticLlama]

http://imgur.com/mceBq"

I'm curious to see if my conclusion is correct.

thanks in advance.

 

[CompuChip]

I don't really agree with your claim that if for two vectors a (non-zero) and b, axb = 0, then b = 0.

 

[Mark44]

Following up with what CompuChip said, what is a X 2a?

 

[ChaoticLlama]

I don't really agree with your claim that if for two vectors a (non-zero) and b, axb = 0, then b = 0.

I'm not sure why you don't agree. I end up with the statement:

a X (b - c) = 0

And for that statement to hold, the following must necessarily be true:

b - c = 0

since we know that a is nonzero.

Following up with what CompuChip said, what is a X 2a?

a is parallel to 2a, so taking the cross-product of two parallel vectors yields the zero vector, because there are an infinite number of vectors perpendicular to a single line.


Sorry, but I don't seem to be following the logic you are trying to lead me through.

 

[HallsofIvy]

The logic he is trying to lead you through is: Since, as you say, the cross product of two parallel vectors is 0, it does NOT follow that is $u\times v= 0$ then either u or v must be 0! In particular, if a is not the 0 vector, then neither is 2a but, again, $a\times 2a= 0$.

If $a\times (b- c)= 0$ then the best you can say is that b- c is parallel to a: that is, that b- c is a multiple of a.