VectorAngle
✖
VectorAngle
Details

- VectorAngle gives an angle in radians.
- For nonzero real vectors the vector angle
satisfies
.
- For complex vectors the numerator is
.
Examples
open allclose allBasic Examples (2)Summary of the most common use cases
The angle between two vectors in 2D:

https://wolfram.com/xid/0cg57h6j6-nntcdf


https://wolfram.com/xid/0cg57h6j6-d60l59

The angle between two vectors in 3D:

https://wolfram.com/xid/0cg57h6j6-dgnv16


https://wolfram.com/xid/0cg57h6j6-cnl6ay

The angle between orthogonal vectors:

https://wolfram.com/xid/0cg57h6j6-fa9z0v


https://wolfram.com/xid/0cg57h6j6-i5t74u


https://wolfram.com/xid/0cg57h6j6-doa4m2

Scope (2)Survey of the scope of standard use cases
Use exact arithmetic to compute the vector angle:

https://wolfram.com/xid/0cg57h6j6-t8hev

https://wolfram.com/xid/0cg57h6j6-cgif0r


https://wolfram.com/xid/0cg57h6j6-cbz6jf

Use 47-digit precision arithmetic:

https://wolfram.com/xid/0cg57h6j6-dhtrv


https://wolfram.com/xid/0cg57h6j6-c6nqax


https://wolfram.com/xid/0cg57h6j6-caxuyp


https://wolfram.com/xid/0cg57h6j6-sh65l

Generalizations & Extensions (1)Generalized and extended use cases
Applications (3)Sample problems that can be solved with this function
Find when two vectors have the same direction:

https://wolfram.com/xid/0cg57h6j6-dz4ged

https://wolfram.com/xid/0cg57h6j6-mxoyer


https://wolfram.com/xid/0cg57h6j6-gtl0

Find the area of the triangle, with u and v as two sides:

https://wolfram.com/xid/0cg57h6j6-dm2vy

https://wolfram.com/xid/0cg57h6j6-bpp0ed

Plot the area in the triangle formed by the axis and a unit vector in the first quadrant:

https://wolfram.com/xid/0cg57h6j6-kk5s5a

Distribution of angles between random vectors with positive entries in 2, 3, 5, and 10 dimensions:

https://wolfram.com/xid/0cg57h6j6-v8i5i

Properties & Relations (6)Properties of the function, and connections to other functions

https://wolfram.com/xid/0cg57h6j6-ek4mu5

https://wolfram.com/xid/0cg57h6j6-cdd98i


https://wolfram.com/xid/0cg57h6j6-co9v5m

The generalization to complex vectors satisfies :

https://wolfram.com/xid/0cg57h6j6-er5c87

https://wolfram.com/xid/0cg57h6j6-fkbi3j


https://wolfram.com/xid/0cg57h6j6-haa8i6

If you rotate a vector u in a plane that includes u, then the vector angle is the rotation angle:

https://wolfram.com/xid/0cg57h6j6-lduaq1

https://wolfram.com/xid/0cg57h6j6-echb2u


https://wolfram.com/xid/0cg57h6j6-taxit

If you rotate it in a plane that does not include u, then the angles differ:

https://wolfram.com/xid/0cg57h6j6-m22v1


https://wolfram.com/xid/0cg57h6j6-jww1w6

The vector angle is related to the cross product through
:

https://wolfram.com/xid/0cg57h6j6-cq2cq7


https://wolfram.com/xid/0cg57h6j6-n6954


https://wolfram.com/xid/0cg57h6j6-ciney7

ArcTan of two arguments gives the signed vector angle between the axis and the vector:

https://wolfram.com/xid/0cg57h6j6-fykgrd

https://wolfram.com/xid/0cg57h6j6-gk9si


https://wolfram.com/xid/0cg57h6j6-dh7r9e


https://wolfram.com/xid/0cg57h6j6-blol34

Eigenvectors are the vectors for which the angle between and
is 0:

https://wolfram.com/xid/0cg57h6j6-inxeev

https://wolfram.com/xid/0cg57h6j6-jrzv4

https://wolfram.com/xid/0cg57h6j6-ioldpk


https://wolfram.com/xid/0cg57h6j6-jwqeba


https://wolfram.com/xid/0cg57h6j6-gsjjfb


https://wolfram.com/xid/0cg57h6j6-dzzl28

Wolfram Research (2007), VectorAngle, Wolfram Language function, https://reference.wolfram.com/language/ref/VectorAngle.html.
Text
Wolfram Research (2007), VectorAngle, Wolfram Language function, https://reference.wolfram.com/language/ref/VectorAngle.html.
Wolfram Research (2007), VectorAngle, Wolfram Language function, https://reference.wolfram.com/language/ref/VectorAngle.html.
CMS
Wolfram Language. 2007. "VectorAngle." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/VectorAngle.html.
Wolfram Language. 2007. "VectorAngle." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/VectorAngle.html.
APA
Wolfram Language. (2007). VectorAngle. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VectorAngle.html
Wolfram Language. (2007). VectorAngle. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VectorAngle.html
BibTeX
@misc{reference.wolfram_2025_vectorangle, author="Wolfram Research", title="{VectorAngle}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/VectorAngle.html}", note=[Accessed: 29-March-2025
]}
BibLaTeX
@online{reference.wolfram_2025_vectorangle, organization={Wolfram Research}, title={VectorAngle}, year={2007}, url={https://reference.wolfram.com/language/ref/VectorAngle.html}, note=[Accessed: 29-March-2025
]}