I'm curious to see if my conclusion is correct.
thanks in advance.
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?
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.
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 [itex]u\times v= 0[/itex] then either u or v must be 0! In particular, if a is not the 0 vector, then neither is 2a but, again, [itex]a\times 2a= 0[/itex].
If [itex]a\times (b- c)= 0[/itex] 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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