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Calculating an angle from vectors

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The dot product of two vectors is the cosine of the angle between them times the magnitudes of the vectors. So, if you have two vectors A <x, y> and B <z, w> with an angle T between them, then:

dot(A, B) = cos(T)*mag(A)*mag(B)

so

T = arccos(dot(A,B)/(mag(A)*mag(B))

with
dot(A,B) = x*z + y*w
mag(A) = sqrt(x*x + y*y)
mag(B) = sqrt(z*z + w*w)

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I might not entirely understand what you're asking for. If I don't, then I apologize.

To find the angle between two vectors A and B, use the geometric formula for the dot product:


A dot B = |A| * |B| * cos(alpha)


where alpha is the angle in question. With some manipulations, we have:


cos(alpha) = (A dot B) / (|A| * |B|)
alpha = cos-1((A dot B) / (|A| * |B|))

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