Can the Dot Product of Two Unit Vectors Ever Be Less Than 1?

onako
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I need certain stopping criterion for approximating one unit vector with another. In case there is a perfect match (after a number of iterations), the dot product of the vectors is 1. I need to know (and have a reasoning for) whether in any other case the dot product of the original unit vector and the approximation (which is also a unit vector) is less than 1.
Thanks
 
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a.b=|a| |b| cos(a,b)

If |a|=1, |b|=1, then a.b=cos(a,b) <= 1 and =1 if and only if a=b.
 
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Thanks.
 

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