Two definitions for a reference frame

Question

My textbook defines a reference frame in two different ways:

1. A collection of at least 3 collinear points that are rigidly connected.

2. A reference frame is defined by three orthogonal unit vectors and one point (the origin).

These two seem like two completely different definitions to me. Could someone explain what the connection is between these two definitions? The textbook attempts to explain this by writing "a reference frame is equivalent to a rigid body. Since all points of a rigid body are fixed with respect to one another, we can use them to define a reference frame", but I am still having trouble understanding the connection.

Answer 1

If you have 3 different points on rigid body you can create orthonormal coordinate system with those equations

$$\vec Z(t)=\dfrac{\overrightarrow{R}_{13}\times \overrightarrow{R}_{12}}{\left| \overrightarrow{R}_{13}\times \overrightarrow{R}_{12}\right| }$$

$$\vec Y(t)=\dfrac{\overrightarrow{R}12}{\left| \overrightarrow{R}_{12}\right| }$$

$$ \overrightarrow{X}(t)=\vec Y\times \vec Z$$

Thus the Transformation matrix between body system and inertial system is

$$R=[\vec X, \vec Y, \vec Z]$$

with $R^T\,R=I_3$

But you also can choose 3 euler angles to create orthogonal transformation matrix R

Thus

$$R=R\left( \alpha ,\beta ,\gamma\right) $$

Edit

$$\overrightarrow{R}_{13}=\overrightarrow{R}_{3}\left( t\right) -\overrightarrow{R}_{1}\left( t\right) $$

$$\overrightarrow{R}_{12}=\overrightarrow{R}_{2}\left( t\right) -\overrightarrow{R}_{1}\left( t\right) $$

Answer 2

Sorry to say, but I see no connection between those two definitions. All $N$ collinear points in a rigid body by definition gives only 1 same line. However, three orthogonal vectors with origin are semantically equal to 4 non-collinear points forming vertexes of Tetrahedron :