Editing Rotation matrix (section)

===General 3D rotations===

Other 3D rotation matrices can be obtained from these three using [[matrix multiplication]].  For example, the product
:<math>\begin{align}
  R = R_z(\alpha) \, R_y(\beta) \, R_x(\gamma) &=
  \overset\text{yaw}
  {\begin{bmatrix}
    \cos \alpha & -\sin \alpha & 0 \\
    \sin \alpha &  \cos \alpha & 0 \\
              0 &            0 & 1 \\
  \end{bmatrix}}
  \overset\text{pitch}
  {\begin{bmatrix}
     \cos \beta & 0 & \sin \beta \\
              0 & 1 &          0 \\
    -\sin \beta & 0 & \cos \beta \\
  \end{bmatrix}}
  \overset\text{roll}
  {\begin{bmatrix}
    1 &  0          &            0 \\
    0 & \cos \gamma & -\sin \gamma \\
    0 & \sin \gamma &  \cos \gamma \\
  \end{bmatrix}} \\
  &= \begin{bmatrix}
        \cos\alpha\cos\beta &
          \cos\alpha\sin\beta\sin\gamma - \sin\alpha\cos\gamma &
          \cos\alpha\sin\beta\cos\gamma + \sin\alpha\sin\gamma \\
        \sin\alpha\cos\beta &
          \sin\alpha\sin\beta\sin\gamma + \cos\alpha\cos\gamma &
          \sin\alpha\sin\beta\cos\gamma - \cos\alpha\sin\gamma \\
       -\sin\beta & \cos\beta\sin\gamma & \cos\beta\cos\gamma \\
  \end{bmatrix}
\end{align}</math>

represents a rotation whose [[yaw, pitch, and roll]] angles are {{mvar|α}}, {{mvar|β}} and {{mvar|γ}}, respectively. More formally, it is an [[Euler angles#Conventions by intrinsic rotations|intrinsic rotation]] whose [[Tait–Bryan angles]] are {{mvar|α}}, {{mvar|β}}, {{mvar|γ}}, about axes {{mvar|z}}, {{mvar|y}}, {{mvar|x}}, respectively.
Similarly, the product
:<math>\begin{align} \\
  R = R_x(\gamma) \, R_y(\beta) \, R_z(\alpha) &=
  \overset\text{roll}
  {\begin{bmatrix}
    \cos \gamma & -\sin \gamma & 0 \\
    \sin \gamma &  \cos \gamma & 0 \\
     0          &   0          & 1 \\
  \end{bmatrix}}
  \overset\text{pitch}
  {\begin{bmatrix}
    \cos \beta & 0 & \sin \beta \\
     0         & 1 &  0 \\
   -\sin \beta & 0 & \cos \beta \\
  \end{bmatrix}}
  \overset\text{yaw}
  {\begin{bmatrix}
    1 &  0          &   0 \\
    0 & \cos \alpha & -\sin \alpha \\
    0 & \sin \alpha &  \cos \alpha \\
  \end{bmatrix}} \\
  &= \begin{bmatrix}
    \cos\beta\cos\gamma & \sin\alpha\sin\beta\cos\gamma - \cos\alpha\sin\gamma & \cos\alpha\sin\beta\cos\gamma + \sin\alpha\sin\gamma \\
    \cos\beta\sin\gamma & \sin\alpha\sin\beta\sin\gamma + \cos\alpha\cos\gamma & \cos\alpha\sin\beta\sin\gamma - \sin\alpha\cos\gamma \\
    -\sin\beta          & \sin\alpha\cos\beta                                  & \cos\alpha\cos\beta \\
  \end{bmatrix} 
\end{align}</math>
represents an [[Euler angles#Conventions by extrinsic rotations|extrinsic rotation]] whose (improper) [[Euler angles]] are {{mvar|α}}, {{mvar|β}}, {{mvar|γ}}, about axes {{mvar|x}}, {{mvar|y}}, {{mvar|z}}.

These matrices produce the desired effect only if they are used to premultiply [[column vector]]s, and (since in general matrix multiplication is not [[commutative]]) only if they are applied in the specified order (see [[#Ambiguities|Ambiguities]] for more details). The order of rotation operations is from right to left; the matrix adjacent to the column vector is the first to be applied, and then the one to the left.<ref>{{cite web |title=Rotation Matrices |url=http://extranet.nmrfam.wisc.edu/nmrfam_documents/bchm800/notes/chapt4.pdf |access-date=30 November 2021}}</ref>