While I was solving a problem something interesting came up if we know the dot product of 2 vectors, and one of the vectors is known. Can we find the other? The interesting bit about this is that even though the dot product behaves pretty much like normal algebraic multiplication 1, but that does not extend to properties like division for example if $a * b = c$ then $a = \frac{b}{c}$
Why is this so (other than the fact that we're talking about vectors)?
1 As Qiaochu Yuan points out that statement is simply inaccurate. Sorry.