# Differences Between Algebraic Multiplication & The Dot Product

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.

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Well, the dot product of two vectors is a scalar, not a vector, so you get much less information out of a dot product than an ordinary product. (Following this train of thought will lead you to a counterexample pretty quickly.) Also, since the dot product of two vectors is a scalar, it doesn't make sense to talk about the dot product of more than two vectors, so the dot product is not associative; that's a major way in which it does not behave "pretty much like normal algebraic manipulation." – Qiaochu Yuan Feb 4 '11 at 15:57

The reason why the dot product of two vectors has no inverse is that it is not an injective function. In fact, if you fix a vector $v$ and a number $r$, then the set of all vectors $w$ that satisfy $v * w = r$ is a hyperplane of your vector space, and those usually contain more than one element.