Coordinate system transformation (6.93-6.97)

Given two coordinate systems $O, \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ and $O^{\prime }, \mathbf{e}_{1}^{\prime }$, $\mathbf{e}_{2}^{\prime }, \mathbf{e}_{3}^{\prime }$. The origin of the second coordinate system has coordinates $a_{10}, a_{20}, a_{30}$ in the first system, and the basis vectors of the second system have coordinates $\left(a_{11}, a_{21}, a_{31}\right)$, $\left(a_{12}, a_{22}, a_{32}\right), \left(a_{13}, a_{23}, a_{33}\right)$ respectively in the basis of the first system. In the first coordinate system a plane is given by the equation $Ax + By + Cz + D = 0$. Find the equation of this plane in the second system.

In space, four points $A(1, 2, 1)$, $B(- 1, 3, 0), C(2, 5, 3), D(- 2, 3, 4)$ and a plane $2x + y - 3z + 2 = 0$ are given. Find the equation of this plane in the new coordinate system $A, \overrightarrow {AB}, \overrightarrow {AC}, \overrightarrow {AD}$.

The planes $x - 2y + 3z - 6 = 0, 2x + y - z = 0, 4x + z - 5 = 0$ are respectively the planes $O^{\prime } y^{\prime } z^{\prime }, O^{\prime } z^{\prime } x^{\prime }$, $O^{\prime } x^{\prime } y^{\prime }$ of a new coordinate system, and the point $A(2, 0, 1)$ has coordinates $1, 1, 1$ in the new system.

In a rectangular coordinate system $O, \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ a plane is given by the equation $3x + 5y + \sqrt{2} z + \sqrt{2} = 0$. The origin of the new rectangular coordinate system is at the point $O^{\prime }(1, 1, - 1)$, the basis vector $\mathbf{e}_{3}^{\prime }$ is opposite to the vector $\mathbf{e}_{3}$, and the basis vectors $\mathbf{e}_{1}^{\prime }$ and $\mathbf{e}_{2}^{\prime }$ are obtained from the vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ respectively by a rotation, in the plane containing $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$, through an angle of $45^{\circ }$ in the direction of the shortest rotation from $\mathbf{e}_{1}$ to $\mathbf{e}_{2}$.

Three planes, given in a rectangular coordinate system by the equations $x + 2y - 2z + 3 = 0, 2x + y + 2z = 0$, $2x - 2y - z + 3 = 0$ (verify that they are pairwise perpendicular), are respectively the planes $O^{\prime } y^{\prime } z^{\prime }, O^{\prime } z^{\prime } x^{\prime }$, $O^{\prime } x^{\prime } y^{\prime }$ of a new rectangular coordinate system, and the point $A(- 1, 0, 0)$ has positive coordinates in the new system.