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AngleVector[θ]

gives the list representing the 2D unit vector at angle θ relative to the axis.

AngleVector[{r,θ}]

gives the list representing the 2D vector of length r at angle θ.

AngleVector[{x,y},θ]

gives the result of starting from the point {x,y}, then going a unit distance at angle θ.

AngleVector[{x,y},{r,θ}]

gives the result of starting from the point {x,y}, then going distance r at angle θ.

Details
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AngleVector

AngleVector[θ]

gives the list representing the 2D unit vector at angle θ relative to the axis.

AngleVector[{r,θ}]

gives the list representing the 2D vector of length r at angle θ.

AngleVector[{x,y},θ]

gives the result of starting from the point {x,y}, then going a unit distance at angle θ.

AngleVector[{x,y},{r,θ}]

gives the result of starting from the point {x,y}, then going distance r at angle θ.

Details

  • Unless explicitly given as a Quantity object, the angle θ is assumed to be in radians, counterclockwise starting from the axis. (Multiply by Degree to convert from degrees.)
  • AngleVector[{r,θ}] gives a vector that starts at {0,0}.
  • The arguments of AngleVector can be symbolic. They can also be Quantity objects.

Examples

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Basic Examples  (4)

Unit vector at angle θ:

Wolfram Language code: AngleVector[θ]

Angles are assumed to be in radians:

Wolfram Language code: AngleVector[Pi / 6]

Unit vector at angle 30°:

Wolfram Language code: AngleVector[30Degree]

Use a Quantity angle instead:

Wolfram Language code: AngleVector[Quantity[30, "AngularDegrees"]]

General symbolic case:

Wolfram Language code: AngleVector[{x, y}, {r, θ}]

Scope  (5)

Unit vector at angle Pi/3:

Wolfram Language code: AngleVector[Pi / 3]

Use degrees:

Wolfram Language code: AngleVector[60Degree]

Use Quantity angles in input:

Wolfram Language code: AngleVector[Quantity[Pi / 3, "Radians"]]
Wolfram Language code: AngleVector[Quantity[60, "AngularDegrees"]]

Specify the norm of the vector:

Wolfram Language code: AngleVector[{5, Pi / 4}]

Specify the origin of the vector:

Wolfram Language code: AngleVector[{a, b}, {5, Pi / 4}]

Use Quantity values in input:

Wolfram Language code: AngleVector[Quantity[{1, 2}, "Meters" / "Seconds"], {Quantity[2, "Miles" / "Hours"], Quantity[45.8, "AngularDegrees"]}]

Properties & Relations  (4)

Reconstruct input:

Wolfram Language code: AngleVector[{3 / 2, 2Pi / 3}]
Wolfram Language code: {Norm[%], Apply[ArcTan, %]}

For short displacements around a geo location, AngleVector approximates GeoDestination:

Wolfram Language code: p = GeoPosition[["London"]]

Move 10 kilometers with initial bearing of 40°:

Wolfram Language code: GeoDestination[p, {10000., 40.}]

Re-express that position in terms of a Cartesian frame centered at London:

Wolfram Language code: GeoPositionENU[%, p]

The horizontal displacement vector in meters is approximately the following:

Wolfram Language code: AngleVector[{10000., (90 - 40) Degree}]

Folding of AngleVector can be used to move from a point along a sequence of {r,θ} displacements:

Wolfram Language code: points = FoldList[AngleVector, {0, 0}, Table[{i, i Pi / 4}, {i, 10}]]
Wolfram Language code: Graphics[Line[points]]

The same result can be achieved with AnglePath, using angles relative to the previous segment:

Wolfram Language code: points === AnglePath[{0, 0}, Table[{i, Pi / 4}, {i, 10}]]

ListPolarPlot can be interpreted as a combination of AngleVector and ListPlot:

Wolfram Language code: pairs = Table[{i 2Pi / 100, Sqrt[i]}, {i, 100}];
Wolfram Language code: ListPolarPlot[pairs]

ListPolarPlot takes pairs {θi,ri}, but AngleVector takes pairs {r,θ}, so Reverse is needed:

Wolfram Language code: ListPlot[AngleVector /@ Reverse /@ pairs, AspectRatio -> Automatic]
Wolfram Research (2015), AngleVector, Wolfram Language function, https://reference.wolfram.com/language/ref/AngleVector.html.

Text

Wolfram Research (2015), AngleVector, Wolfram Language function, https://reference.wolfram.com/language/ref/AngleVector.html.

CMS

Wolfram Language. 2015. "AngleVector." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AngleVector.html.

APA

Wolfram Language. (2015). AngleVector. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AngleVector.html

BibTeX

@misc{reference.wolfram_2026_anglevector, author="Wolfram Research", title="{AngleVector}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/AngleVector.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_anglevector, organization={Wolfram Research}, title={AngleVector}, year={2015}, url={https://reference.wolfram.com/language/ref/AngleVector.html}, note=[Accessed: 12-June-2026]}