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A.B=A.C Then B=C True or false? If true, prove it in general terms, if false, provide a counter-example.

Ok, I just need some body to comment on my little proof here, and any guidelines to make it more thorough or whatnot.

I know that the dot product is commutative,

A.(B+C)=A.B +A.C, but not sure if it really needs to be in my proof or not.

Proof

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Say A.B=N and A.C=N (where N is a scalar number)

so if N=N

Then A.B=A.C

If I cancel the A's, I get B=C.

Is that a good way to approach that, or is there a better way of expressing it?