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I read that given a vector space $V$, and two vectors in $V$, then the two vectors may be orthogonal under one inner product definition but not orthogonal under a different inner product. Considering that, would it be fair to say that one of the purposes of defining an inner product is to define what it means for two vectors to be perpendicular?

Separately, are there any definitions of inner product other than the standard Euclidean inner product in $\mathbb{R}^n$ that are widely used in practical applications? I guess that would that be equivalent to asking if there are alternative definitions of orthoginality which are practically useful?

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    I always think of the inner product intuitively as measuring "angles between vectors", although this is not a valid interpretation in every case. So not only does it determine orthogonality, it determines the range of angles.2012-06-27
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    There is a subtle difference between fixing an inner product and fixing a notion of orthogonality: the latter does not "see" a positive scalar factor by which the inner product might be multiplied.2012-06-27
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    @MarcvanLeeuwen You mean like a weighted Euclidean inner product?2012-06-27

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