This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# VectorAngle

 VectorAngle gives the angle between the vectors u and v.
• For nonzero real vectors the vector angle satisfies .
• For complex vectors the numerator is .
The angle between two vectors in 2D:
The angle between two vectors in 3D:
The angle between orthogonal vectors:
The angle between two vectors in 2D:
 Out[1]=
 Out[2]=
The angle between two vectors in 3D:
 Out[3]=
 Out[4]=

The angle between orthogonal vectors:
 Out[1]=
 Out[2]=
 Out[3]=
 Scope   (2)
Use exact arithmetic to compute the vector angle:
Use machine arithmetic:
Use 47-digit precision arithmetic:
Use symbolic vectors:
For complex vectors the angle returned may be complex:
 Applications   (3)
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:
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:
The angle between the zero vector and any other vector is indeterminate:
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