({{, (RefLink[AngleVector, paclet:ref/AngleVector])[θ] 
gives the list representing the 2D unit vector at angle θ relative to the x axis.}, {, (RefLink[AngleVector, paclet:ref/AngleVector])[{r,θ}]
gives the list representing the 2D vector of length r at angle θ.}, {, (RefLink[AngleVector, paclet:ref/AngleVector])[{x,y},θ]
gives the result of starting from the point {x,y}, then going a unit distance at angle θ.}, {, (RefLink[AngleVector, paclet:ref/AngleVector])[{x,y},{r,θ}]
gives the result of starting from the point {x,y}, then going distance r at angle θ.}})

OpenersNotesSection(Spacer1)Details

Unless explicitly given as a (RefLink[Quantity, paclet:ref/Quantity]) object, the angle θ is assumed to be in radians, counterclockwise starting from the x axis. (Multiply by (RefLink[Degree, paclet:ref/Degree]) to convert from degrees.)

(RefLink[AngleVector, paclet:ref/AngleVector])[{r,θ}] gives a vector that starts at {0,0}.

The arguments of (RefLink[AngleVector, paclet:ref/AngleVector]) can be symbolic. They can also be (RefLink[Quantity, paclet:ref/Quantity]) objects.

OpenersPrimaryExamplesSection(Spacer1)Examples  (13)

OpenersExampleSection0(Spacer1)Basic Examples  (4)

Unit vector at angle θ:

AngleVector[θ]

Angles are assumed to be in radians:

AngleVector[Pi/6]

Unit vector at angle 30°:

AngleVector[30Degree]

Use a (RefLink[Quantity, paclet:ref/Quantity]) angle instead:

AngleVector[Quantity[30,"AngularDegrees"]]

General symbolic case:

AngleVector[{x,y},{r,θ}]

OpenersExampleSection1(Spacer1)Scope  (5)

Unit vector at angle (RefLink[Pi, paclet:ref/Pi])/3:

AngleVector[Pi/3]

Use degrees:

AngleVector[60Degree]

Use (RefLink[Quantity, paclet:ref/Quantity]) angles in input:

AngleVector[Quantity[Pi/3,"Radians"]]

AngleVector[Quantity[60,"AngularDegrees"]]

Specify the norm of the vector:

AngleVector[{5,Pi/4}]

Specify the origin of the vector:

AngleVector[{a,b},{5,Pi/4}]

Use (RefLink[Quantity, paclet:ref/Quantity]) values in input:

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

OpenersExampleSection3(Spacer1)Properties & Relations  (4)

Reconstruct input:

AngleVector[{3/2,2Pi/3}]

{Norm[%],Apply[ArcTan,%]}

For short displacements around a geo location, (RefLink[AngleVector, paclet:ref/AngleVector]) approximates (RefLink[GeoDestination, paclet:ref/GeoDestination]):

p=GeoPosition[LinguisticAssistant]

Move 10 kilometers with initial bearing of 40°:

GeoDestination[p,{10000.,40.}]

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

GeoPositionENU[%,p]

The horizontal displacement vector in meters is approximately the following:

AngleVector[{10000.,(90-40) Degree}]

Folding of (RefLink[AngleVector, paclet:ref/AngleVector]) can be used to move from a point along a sequence of {r,θ} displacements:

points=FoldList[AngleVector,{0,0},Table[{i,i Pi/4},{i,10}]]

Graphics[Line[points]]

The same result can be achieved with (RefLink[AnglePath, paclet:ref/AnglePath]), using angles relative to the previous segment:

points===AnglePath[{0,0},Table[{i,Pi/4},{i,10}]]

(RefLink[ListPolarPlot, paclet:ref/ListPolarPlot]) can be interpreted as a combination of (RefLink[AngleVector, paclet:ref/AngleVector]) and (RefLink[ListPlot, paclet:ref/ListPlot]):

pairs=Table[{i 2Pi/100,Sqrt[i]},{i,100}];

ListPolarPlot[pairs]

(RefLink[ListPolarPlot, paclet:ref/ListPolarPlot]) takes pairs {(Subscript[θ, i]),(Subscript[r, i])}, but (RefLink[AngleVector, paclet:ref/AngleVector]) takes pairs {r,θ}, so (RefLink[Reverse, paclet:ref/Reverse]) is needed:

ListPlot[AngleVector/@Reverse/@pairs,AspectRatio->Automatic]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[AnglePath, paclet:ref/AnglePath])  ▪  (RefLink[FromPolarCoordinates, paclet:ref/FromPolarCoordinates])  ▪  (RefLink[Sin, paclet:ref/Sin])  ▪  (RefLink[Cos, paclet:ref/Cos])  ▪  (RefLink[Degree, paclet:ref/Degree])  ▪  (RefLink[FromDMS, paclet:ref/FromDMS])  ▪  (RefLink[AbsArg, paclet:ref/AbsArg])  ▪  (RefLink[UnitVector, paclet:ref/UnitVector])  ▪  (RefLink[Rotate, paclet:ref/Rotate])  ▪  (RefLink[VectorAngle, paclet:ref/VectorAngle])  ▪  (RefLink[CirclePoints, paclet:ref/CirclePoints])  ▪  (RefLink[GeoDestination, paclet:ref/GeoDestination])  ▪  (RefLink[ListPolarPlot, paclet:ref/ListPolarPlot])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Trigonometric Functions, paclet:guide/TrigonometricFunctions])}, {▪(RefLinkPlain[Angles and Polar Coordinates, paclet:guide/AnglesAndPolarCoordinates])}, {▪(RefLinkPlain[Operations on Vectors, paclet:guide/OperationsOnVectors])}})}})