Rotation matrix for rotations around y-axis
R = roty(ang)
Construct the matrix for a rotation of a vector around the y-axis by 45°. Then let the matrix operate on a vector.
R = roty(45)
R = 0.7071 0 0.7071 0 1.0000 0 -0.7071 0 0.7071
v = [1;-2;4]; y = R*v
y = 3.5355 -2.0000 2.1213
Under a rotation around the y-axis, the y-component of a vector is invariant.
ang— Rotation angle
Rotation angle specified as a real-valued scalar. The rotation angle is positive if the rotation is in the counter-clockwise direction when viewed by an observer looking along the y-axis towards the origin. Angle units are in degrees.
R— Rotation matrix
3-by-3 rotation matrix returned as
Rotation matrices are used to rotate a vector into a new direction.
In transforming vectors in three-dimensional space, rotation matrices are often encountered. Rotation matrices are used in two senses: they can be used to rotate a vector into a new position or they can be used to rotate a coordinate basis (or coordinate system) into a new one. In this case, the vector is left alone but its components in the new basis will be different from those in the original basis. In Euclidean space, there are three basic rotations: one each around the x, y and z axes. Each rotation is specified by an angle of rotation. The rotation angle is defined to be positive for a rotation that is counterclockwise when viewed by an observer looking along the rotation axis towards the origin. Any arbitrary rotation can be composed of a combination of these three (Euler’s rotation theorem). For example, one can rotated a vector using a sequence of three rotations: .
The rotation matrices that rotate a vector around the x, y, and z-axes are given by:
Counterclockwise rotation around x-axis
Counterclockwise rotation around y-axis
Counterclockwise rotation around z-axis
The following three figures show what positive rotations look like for each rotation axis:
For any rotation, there is an inverse rotation satisfying . For example, the inverse of the x-axis rotation matrix is obtained by changing the sign of the angle:
We can think of rotations in another way. Consider the original set of basis vectors, , and rotate them all using the rotation matrix A. This produces a new set of basis vectors related to the original by:
Now any vector can be written as a linear combination of either set of basis vectors:
 Goldstein, H., C. Poole and J. Safko, Classical Mechanics, 3rd Edition, San Francisco: Addison Wesley, 2002, pp. 142–144.
Usage notes and limitations:
Does not support variable-size inputs.