The orientation of body in space is defined 3 by angles. The step by step rotation process and chain of three-dots multiplication give an easy way to compute pile of rotations in 3D and high dimensional space and give a general orientation system. A visualization of quaternion is proposed.

The orientation of a rigid body in space is defined by 3 angles, for example the angles of pitch, roll and yaw of an airplane. The most commonly used orientation systems are Euler angles and Tait–Bryan angles. Quaternion is often used for computing rotation of body.

In this article we will present a new system for computing rotation in space that consists of rotating the proper frame of reference of the rigid body with respect to that of the space. This system is more general than Euler angles and Tait–Bryan angles and easier to use than rotation matrix and quaternion.

Below we will refer 2-dimensional space as 2D and 3-dimensional space as 3D. First, we will explain our approach using 2D body, then apply it to 3D body and body of even higher dimensions. We will use our system to give a visualization of the geometrical meaning of quaternion in 3D space.

Let us begin with defining the frame of reference of a rotated body and that of the space. A frame is defined by its basis that is a set of orthogonal unit vectors. The frame of the space is fixed. The frame of the rotated body is solidary to the body and rotates with it. The orientation in space of the body is defined by the angles its frame makes with that of the space.

We refer the basis of the frame of the space as space
basis and that of the body the proper basis. The
set of unit vectors of the space basis is (**e**_{x},** e**_{y}) for 2D space and (**e**_{x},** e**_{y},** e**_{z}) for 3D space.
The set of unit vectors of the proper basis is (**e**_{x}^{i},** e**_{y}^{i}) and (**e**_{x}^{i},** e**_{y}^{i},** e**_{z}^{i}) when it has gone *i* rotations. *i=0* at the
start.

…

Figures and equations are in the article below:

«Step by step rotation in normal and high dimensional space and meaning of quaternion»

https://pengkuanonmaths.blogspot.com/2021/09/step-by-step-rotation-in-normal-and.html