Are there physical quantities constitute of magnitude, direction and rotation along that direction?

Question

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:

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

Answer 1

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 :

$$\vec{v}_I=R\,\vec{v}_B$$

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

$$R_{BI}=R_3(\psi)\,R_2(\vartheta)\,R_1(\varphi)$$

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