How this calculator works
Cosine similarity measures alignment, independently of positive rescaling. It is 1 for equal directions, 0 for orthogonal vectors and -1 for opposite directions. Components may be negative, so the result is not a percentage. The computation normalises each vector before taking its dot product to reduce unnecessary overflow.
Formula
s = (A dot B)/(|A||B|); cosine distance = 1 - s; angle = arccos(s).
Worked example
For A = (1, 0) and B = (1, 1), the similarity is 1/sqrt(2), about 0.7071067812, and the angle is 45 degrees. Opposite unit vectors have similarity -1.
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Frequently asked questions
Why reject a zero vector?
A zero vector has no direction, and the defining denominator is zero.
Does similarity imply equal magnitudes?
No. Vectors (1, 2) and (10, 20) have similarity 1 despite their different lengths.
Is cosine distance a metric?
In general 1 minus cosine similarity does not satisfy the triangle inequality. Angular distance is a different construction.