Lin Alg Question: Cross-Product Proof

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The discussion centers on the properties of the cross product of vectors, specifically addressing the claim that if the cross product of two vectors a and b equals zero, then one of the vectors must be zero. Participants argue that if a is non-zero and a × b = 0, it implies that b is parallel to a, not necessarily that b is zero. The example of a × 2a is used to illustrate that two parallel vectors result in a zero cross product, reinforcing that non-zero vectors can still yield a zero result. Ultimately, the conclusion drawn is that the condition a × (b - c) = 0 indicates that b - c is parallel to a, rather than asserting that either vector must be zero. This highlights the nuanced understanding of vector relationships in linear algebra.
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http://imgur.com/mceBq"

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

thanks in advance.
 
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I don't really agree with your claim that if for two vectors a (non-zero) and b, axb = 0, then b = 0.
 
Following up with what CompuChip said, what is a X 2a?
 
CompuChip said:
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.

Mark44 said:
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.
 
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.
 

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