Matrix R is Rotation Matrix if and only if its determinantdet(R) = 1, its rows and columns are orthonormal and R-1 = RT. If you specify a matrix R that doesn't satisfy those conditions, it will be first transformed in its normalized form, if that is possible.
Quaternion Q is specified as Q (w, x, y, z) = w + x i + y j + z k
If you specify the Quaternion Q which is not unit quaternion (versor), it will be first normalized, if it is possible ( ‖Q‖ ≠ 0 ).
If specifying rotation by Euler angles, you san specify it by either proper Euler angles (XYX, XZX, YXY, YZY,ZXZ, ZYZ) or by Tait-Bryan angles (XYZ, XZY,YXZ, YZX, ZXY, ZYX). By default, rotation angles are intrinsic (internal point of view of the rotating body), but you can also specify them as extrinsic (point of view from the external reference system).
If you just want to convert from one coordinate system ℝ3 to the rotated one ℛ(ℝ3) (column vectors of the Rotation Matrix R are the base vectors of the rotated space expressed in the coordinates of the original space), or from one rotation formalism to another, specify the starting formalism (Rotation Matrix, Quaternio, Axis and Angle or Euler Angles) and press Convert. If, on the other hand, you specify the rotation (fomalism) ℛ and coordinates of a point P ∈ ℝ3, by pressing Rotate button, coordinates of the rotated point PR = ℛ(P) will be calculated, along with the all alternative rotation formalisms.
