§ 3

Derive the formulas for a change of basis and a change of coordinate system on a straight line. How do the coordinates of the points of the line change if, with the origin unchanged, the length of the basis vector is doubled?

Let $O^{\prime }$ be the midpoint of the side $A B$ of the triangle $O A B$. Write the formulas of transition from the coordinate system $O, \overrightarrow {O B}$, $\overrightarrow {O A}$ to the coordinate system $O^{\prime }, \overrightarrow {O^{\prime } O}, \overrightarrow {O^{\prime } B}$.

A Cartesian coordinate system $O, \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ is given. How is the coordinate system $O^{\prime }, \mathbf{e}_{1}^{\prime }$, $\mathbf{e}_{2}^{\prime }, \mathbf{e}_{3}^{\prime }$ situated relative to it if the formulas of transition have the form $x=1-y^{\prime }-z^{\prime }$, $y=1-x^{\prime }-z^{\prime }, z=1-x^{\prime }-y^{\prime }$.

Fig. 4