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The range is     -1.0  ...  1.0

Formula for the Angle between Two Vectors

The range is minus one to plus one, because each dot product in the previous page is:

(1,  0)T · ( cos θ, sin θ)T  =  cos θ

This is true when au is a unit vector pointing in any direction.

The angle between two unit vectors:

au · bu  =  cos θ

where θ is the angle between the two vectors. This formula automatically includes the fact that the dot product of perpendicular vectors is zero (because cos 90° is zero).


(Calculator Problem: ) What is the cosine of the angle between the two unit vectors represented by (0.7071, 0.7071)T and (0.5, 0.866)T ?