Built-in Mathematica Symbol

RotationMatrix

]
gives the 2D rotation matrix that rotates 2D vectors counterclockwise by

radians.

, w]
gives the 3D rotation matrix for a counterclockwise rotation around the 3D vector w.

RotationMatrix[{u, v}]
gives the matrix that rotates the vector u to the direction of the vector v in any dimension.

, {u, v}]
gives the matrix that rotates by

radians in the hyperplane spanned by u and v.

RotationMatrix is set up to use the vector-oriented convention and to give a matrix m so that m.r yields the rotated version of a vector r.

Transpose[RotationMatrix[...]] gives rotation matrices with the alternative coordinate-system-oriented convention for which r.m yields the rotated version of a vector r.

Angles in RotationMatrix are in radians. Degree or ° specifies an angle in degrees.

Positive in RotationMatrix[, {u, v}]corresponds to going from the direction of u towards the direction of v.

RotationMatrix[, w] is equivalent to RotationMatrix[, {u, v}], where uw, vw and u, v, w form a right-handed coordinate system.

RotationMatrix gives an orthogonal matrix of determinant 1, that in n dimensions can be considered an element of the group .

General 2D rotation matrix for rotating a vector about the origin:

Out[1]//MatrixForm=

Counterclockwise rotation by 30°:

Rotation that transforms the direction of

into the direction of

:

3D rotation around the z axis:

Out[1]//MatrixForm=