If vector is represented by coordinates, a=(x1,y1,z1), b=(x2,y2,z2), then a.b=(x1x2+y1y2+z1z2).

|a|=√(x1^2+y1^2+z1^2), |b|=√(x2^2+y2^2+z2^2).

Incorporate these formulas (Ⅰ), receive:

cos < a, b > = (x1x2 + + z1z2 y1y2) / [square root (x1 ^ 2 + y1 z1 ^ ^ 2 + 2) * square root (x2 y2 ^ ^ 2 + 2 + z2 ^ 2)].

The above formulas are given in spatial 3D coordinates, if z=0 in coordinates, the Plane vector Calculator formulas are obtained. The value range of the Angle between two vectors is: [0,π].

When the included Angle is acute, cosθ>0; When the included Angle is obtuse,cosθ<0.