gives the angle between the vectors u and v.
- VectorAngle gives an angle in radians.
- For nonzero real vectors the vector angle satisfies .
- For complex vectors the numerator is .
Examplesopen allclose all
Basic Examples (2)
The angle between two vectors in 2D:
The angle between two vectors in 3D:
The angle between orthogonal vectors:
Use exact arithmetic to compute the vector angle:
Use machine arithmetic:
Use 47-digit precision arithmetic:
Use symbolic vectors:
Generalizations & Extensions (1)
For complex vectors, the angle returned may be complex:
Find when two vectors have the same direction:
Find the area of the triangle, with u and v as two sides:
Plot the area in the triangle formed by the axis and a unit vector in the first quadrant:
Distribution of angles between random vectors with positive entries in 2, 3, 5, and 10 dimensions:
Properties & Relations (6)
The vector angle satisfies :
The generalization to complex vectors satisfies :
If you rotate a vector u in a plane that includes u, then the vector angle is the rotation angle:
If you rotate it in a plane that does not include u, then the angles differ:
The vector angle is related to the cross product through :
ArcTan of two arguments gives the signed vector angle between the axis and the vector:
Eigenvectors are the vectors for which the angle between and is 0:
Possible Issues (1)
The angle between the zero vector and any other vector is indeterminate: