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I want to use Mathematica to show that the inner product of a vector with itself is equal to the square of its norm.

This is what I tried:

$Assumptions = x ∈ Vectors[3, Reals];
expr = Dot[x, x] == Norm[x]^2;

FullSimplify[expr]
(*x.x == Norm[x]^2*)

TensorReduce[expr]
(*x.x == Norm[x]^2*)

I had expected at least one of the last two lines to return True.

Why couldn't Mathematica simplify expr to True in this case? Are there additional assumptions I should include so that it return True?

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    $\begingroup$ One important thing to know is that not every function is supported. I haven't used this functionality much, so I might be wrong, but I think that Norm is simply not (fully) supported. $\endgroup$
    – Szabolcs
    Feb 3, 2016 at 15:46
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    $\begingroup$ The strange thing is that the documentation for Norm explicitly says that For vectors, Norm[v] is Sqrt[v.Conjugate[v]]. expr = Dot[x, x] == Sqrt[Dot[x, Conjugate[x]]]^2 does yield true, so Norm is not considering x to be a vector. $\endgroup$
    – rhermans
    Feb 3, 2016 at 16:17

1 Answer 1

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I think the closest you can get to what you looking for is

Assuming[{a, b, c} ∈ Reals, With[{x = {a, b, c}}, x.x == Norm[x]^2 // Simplify]]

True

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