There are scalar quantities(magnitude) and vector quantities(magnitude and direction), but are there fundamental quantities that also depends on how it's oriented/rotated along the direction(magnitude, direction, and rotation/minor direction) in 3D Euclidean space? Can you give me examples of these quantities?

Just like rigid body rotation, such quantities should be able to described by pitch-roll-yaw as below: enter image description here

As shown, the second minor direction should be orthogonal to the direction (or parallel since it's axial, depends on how you view it), so dyads might not be the answer?

Bonus: since there are not more complex rigid body rotations in 3D, I imagine these quantities are

  1. "fundamental" as vectors, not "derived" as higher order tensors

  2. The "most complex" fundamental quantity in 3D space, otherwise its orientation won't be able to be described with rigid body rotation

  • 3
    $\begingroup$ You want spinors in 3 dimensions or quaternions. $\endgroup$ – Thomas Fritsch Aug 13 at 19:07
  • $\begingroup$ In group theory language, I think you want a geometric object $x$ for which the stabilizer subgroup of $SO(3)$ is trivial. I don't know off the top of my head whether such a thing exists, but I figured I'd throw this out as a comment in case anyone else does. $\endgroup$ – Michael Seifert Aug 14 at 14:00
  • $\begingroup$ @ThomasFritsch Yes, do you by chance know any physical quantities that are spinors or quaternions? $\endgroup$ – lxiangyun93 Aug 15 at 15:15
  • $\begingroup$ In quantum mechanics the wave function $\psi(x)$ of a particle (e.g. an electron) is a spinor field. See Pauli equation. $\endgroup$ – Thomas Fritsch Aug 15 at 15:26

I try to answer your questions:

1) magnitude of a vector.

you have two coordinate systems , one inertial fixed and the other body fixed (airplane fixed).

let say you know the vector components ($\vec{v}_B$) in airplane system (index B) , so the magnitude of velocity vector is:

$$v_1=\sqrt{\vec{v}_B^T\,\vec{v}_B}\tag 1$$

we transfer this vector components to inertial system :


where $R$ is the rotation matrix between body system and inertial system.

if we calculate the amplitude of this vector we get:

$$v_2=\sqrt{\left(R\,\vec{v}_B\right)^T\,R\,\vec{v}_B}=\sqrt{\vec{v}_B^T\,R^T\,R\,\vec{v}_B}=\sqrt{\vec{v}_B^T\,\vec{v}_B}\tag 2$$

with $R^T\,R=I_3\quad \Leftrightarrow v_2=v1$

thus the amplitude of a vector is invariant under orthonormale coordinate transformation (rotation matrix).

2) how to calculate the yaw pitch and roll angles

you can "build" the rotation matrix $R$ out of three elementar rotation matrices:

$$R_1(\alpha)=\left[ \begin {array}{ccc} 1&0&0\\ 0&\cos \left( \alpha \right) &-\sin \left( \alpha \right) \\ 0& \sin \left( \alpha \right) &\cos \left( \alpha \right) \end {array} \right] \quad \text{Rotation about the x-axes}$$

$$R_2(\beta)=\left[ \begin {array}{ccc} \cos \left( \beta \right) &0&\sin \left( \beta \right) \\ 0&1&0\\ -\sin \left( \beta \right) &0&\cos \left( \beta \right) \end {array} \right] \quad \text{Rotation about the y-axes}$$

$$R_3(\gamma)= \left[ \begin {array}{ccc} \cos \left( \gamma \right) &-\sin \left( \gamma \right) &0\\\sin \left( \gamma \right) &\cos \left( \gamma \right) &0\\ 0&0&1\end {array} \right] \quad \text{Rotation about the z-axes}$$

for example $R=R_3(\alpha)\,R_2(\beta)\,R_3(\gamma)$. notice that in this case the first rotation is about the z-axes, the second rotation is about the "new" $y'$-axes and the third is about is about the "new" $z''$-axes. these angles are call euler angles. for each rotation matrix you get singularity (gimbal) in one of the euler angles.

to obtain the yaw ($\psi$) pitch ($\vartheta$) and roll ($\varphi$) angels we choose this rotation matrix between the body system and inertial system


the singularity of this matrix is wenn the pitch angle $\vartheta \pm \pi/2$

  • $\begingroup$ Thanks for your reply! Unfortunately I think it's missing the point somehow, as I was looking for examples of actual physical quantities that can not be described by just magnitude and direction, but magnitude plus pitch-yaw-roll. Do you have any suggestions in mind? @Eli $\endgroup$ – lxiangyun93 Aug 15 at 15:07
  • $\begingroup$ Hey, to answer your question on an answer you deleted, OP means "original poster". i.e. the person who posted the question. $\endgroup$ – Aaron Stevens Aug 16 at 16:44

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