In POV-Ray
predefined vectors

o = < 0, 0, 0> (center of the world!),
x = <1, 0, 0>; y = <0, 1, 0>; z = <0, 0, 1>;
xy = < 1, 1, 0>; yz = < 0, 1, 1>; xz = < 1, 0, 1>;
t = <0, 0, 0, 1>; u = <1, 0>; v = <0, 1>;

Vector addition

<a,b,c> + <d,e,f> = <a+d,b+e,c+f>

Multiplication with scalar values

5 * <1,2,-5> = <5,10,-25>

Extracting components

If V = < a, b, c >, than a = V.x, b = V.y and c = V.z .

Projection on coordinate planes

The projection of P on the xz plane is <P.x,0,P.z>

Length of vectors: |V|

vlength(V)

Dot product V₁*V₂

vdot(V1,V2)

Vector V normated to length 1
a 1 unit vector in the direction of V

vnormalize(V)

Cross product V₁ x V₂

vcross(V1,V2)

Rotation of a vector V₁ around
a vector V₂ by A degrees

vaxis_rotate(V1,V2,A)

Rotation of a vector V₁ around the
axes by V₂.x, V₂.y, V₂.z .

vrotate(V1,V2)

With "math.inc"

Projection of a vector V on
a plane defined by an axis

VProject_Plane(V, Axis)

Projection of a vector V on
the axis

VProject_Axis(V, Axis)

Vector perpendicular to axis
and in the plane of V and axis

VPerp_Adjust(V, Axis)

Vektor perpendicular to V₁ and V₂

VPerp_To_Plane(V1, V2)

Cosine of the angle
between V₁ and V₂

VCos_Angle(V1, V2)

Angle between V₁ and V₂ in radians

VAngle(V1, V2)

Angle between V₁ and V₂ in degrees

VAngleD(V1, V2)

mit "transforms.inc"

Sheering of an object, here
x axis points to A, y to B, z to C.

Shear_Trans(A, B, C)

Sheering of an object like above
additional translation by D.

Matrix_Trans(A, B, C, D)

Stretching along an axis

Axial_Scale_Trans(Axis, Amount)

Rotating around an axis

Axis_Rotate_Trans(Axis, Winkel)

Rotating around a Point
um <Rotx,Roty,Rotz>

Rotate_Around_Trans( <Rotx,Roty,Rotz>, Center_of_Rotation)

Reorientate of Axis1
to new Axis2

Reorient_Trans(Axis1, Axis2)

Reorientate in a way
that y axis points to YAxis

Point_At_Trans(YAxis)