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 .


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Basic Examples  (2)

The angle between two vectors in 2D:

The angle between two vectors in 3D:

The angle between orthogonal vectors:

Scope  (2)

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:

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:

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:

Wolfram Research (2007), VectorAngle, Wolfram Language function,


Wolfram Research (2007), VectorAngle, Wolfram Language function,


Wolfram Language. 2007. "VectorAngle." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2007). VectorAngle. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_vectorangle, author="Wolfram Research", title="{VectorAngle}", year="2007", howpublished="\url{}", note=[Accessed: 20-June-2024 ]}


@online{reference.wolfram_2024_vectorangle, organization={Wolfram Research}, title={VectorAngle}, year={2007}, url={}, note=[Accessed: 20-June-2024 ]}