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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

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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.

In contrast to this, multiplication by a non-zero real number is an injective function (in fact, a bijective function), and so it has an inverse function, namely division by that number.

You should also note that multiplication of real numbers and dot products are fundamentally different in the sense that multiplication of two real numbers gives you back a real number, whereas the dot product of two vectors in general does not give you back a vector of the same space, but a real number (or an element of the field you're working over).

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The meaning of product should not be rather cannot be taken as algebraic multiplication always. The meaning of product actually is the output of the operation performed. For Example: The product(i.e. the output) of the addition operation performed on the real no. 2 and 3 yields 5. The algebraic addition operation is simply an arithmetical tool developed to make the calculation work easier in case any real number is repeatedly added required number of times without bothering about the physics of the problem concerned. On the other hand the dot product of two vectors gives the outcome of an operation applied on the vectors involved by considering the physics of the problem . Hence dot product of two vectors is all together different from their algebraic multiplication (which is not even meaningful).

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