Affine and Orthogonal Changes of Coordinates

Write the formulas for passing from one coordinate system in the plane to another, if the origin of the first system $(x, y)$ is the vertex $A$ of a parallelogram $A B C D$, with basis the vectors $\overrightarrow {A D}, \overrightarrow {A B}$; and the origin of the second system $(x^{\prime }, y^{\prime })$ is the vertex $C$, with basis the vectors $\overrightarrow {C B}, \overrightarrow {C D}$.

Two coordinate systems are given in the plane: $O x y$ and $O^{\prime } x^{\prime } y^{\prime }$. The coordinates $x$ and $y$ of an arbitrary point relative to the first system are expressed in terms of its coordinates $x^{\prime }$ and $y^{\prime }$ relative to the second system by the following formulas: $x = 2 x^{\prime }-5 y^{\prime }+3, y = -x^{\prime }+2 y^{\prime }-2$. Find the coordinates of the origin of the first system and of the unit vectors of its axes relative to the second system.

Two coordinate systems are given in the plane: $O x y, O^{\prime } x^{\prime } y^{\prime }$. The coordinates $(x, y)$ and $(x^{\prime }, y^{\prime })$ of an arbitrary point $M$ relative to these two systems are related by the following transition formulas:

With respect to an affine coordinate system, three points of the plane are given: $A(2,1), B(3,0), C(1,4)$. In a new affine coordinate system, the same points have coordinates $A(1,6), B(1,9), C(3,1)$. Find:

Write the coordinate transformation formulas in the plane, taking as the new axes $O^{\prime } x^{\prime }$ and $O^{\prime } y^{\prime }$ the lines $x-3 y+2 = 0$, $3 x+2 y-1 = 0$, respectively, and as the unit point of the new system the point $(2,3)$.

Write the equations of the line $x-y-5 = 0$ in the coordinate system in the plane whose axes $O^{\prime } x^{\prime }$ and $O^{\prime } y^{\prime }$ are the lines $2 x+y-1 = 0$ and $x-3 y+6 = 0$, respectively, and whose unit point is the point $(-1,0)$.

Through the point $N(2,-3)$ of the plane, draw a line whose segment between the lines $x+4 y+7 = 0,3 x-2 y-4 = 0$ is bisected at the point $N$.

The point $(0,2)$ where the medians of a triangle meet, and the equations of two of its sides, $5 x-4 y+15 = 0,4 x+y-9 = 0$, are given. Find the equation of the third side of the triangle.

The equations of the medians of a triangle, $4 x+5 y = 0, x-3 y = 0$, and its vertex $(2,-5)$, are given. Set up the equations of the sides of the triangle.

Find the formulas for passing from an oblique coordinate system $O x y$ in the plane with coordinate angle $\omega$ to a rectangular system $O x^{\prime } y^{\prime }$, whose axes' positive directions are the bisectors of the first and second coordinate angles of the oblique system, under the condition that the basis vectors of the oblique system have unit length.

Find the formulas for passing from one oblique coordinate system $O x y$ with coordinate angle $\omega$ to another oblique system $O x^{\prime } y^{\prime }$, if the like-named axes of these systems are mutually perpendicular, while the unlike-named axes form acute angles, and all basis vectors of both systems have length 1.

The axes of a rectangular coordinate system in the plane are rotated about the origin through an angle $-\pi / 3$. Find the old coordinates of a point $A$, knowing its coordinates $(4,-2)$ in the new coordinate system.

A transformation of rectangular coordinates in the plane from $O x y$ to $O^{\prime } x^{\prime } y^{\prime }$ is given:

Find the formulas for the inverse transition from the system $O^{\prime } x^{\prime } y^{\prime }$ to $O x y$.

Consider the composition of two transformations of rectangular coordinates in the plane:

A new rectangular coordinate system in the plane is obtained from the old one by translating the origin to the point $O^{\prime }(4,-3)$ and rotating through an angle $\alpha$ such that $\cos \alpha = -4 / 5, \sin \alpha = 3 / 5$. Find the new coordinates of a point $M$, knowing its old coordinates $(5,10)$.

The origin $O^{\prime }$ of a new rectangular system $O^{\prime } x^{\prime } y^{\prime }$ in the plane has coordinates $(-3,-2)$ relative to the rectangular system $O x y$. It is also known that $\cos \left(\widehat{O x, O^{\prime } x^{\prime }}\right) = -\frac{4}{5}, \cos \left(\widehat{O x, O^{\prime } y^{\prime }}\right) = -\frac{3}{5}$ and that the systems $O x y$ and $O^{\prime } x^{\prime } y^{\prime }$ have opposite orientation. Find the expressions of the old coordinates $x$ and $y$ in terms of the new coordinates $x^{\prime }$ and $y^{\prime }$.

As the unit vectors of the first coordinate system in space, three edges $\overrightarrow {O A}, \overrightarrow {O B}, \overrightarrow {O C}$ of a tetrahedron $O A B C$ issuing from one point $O$ are taken, and as the unit vectors of the second system, the medians $\overrightarrow {O D}, \overrightarrow {O E}, \overrightarrow {O F}$ of the faces $B O C, C O A, A O B$ of this tetrahedron. Find the coordinates of the vertices of the tetrahedron in the second system.

Two coordinate systems $O x y z$ and $O^{\prime } x^{\prime } y^{\prime } z^{\prime }$ are given in space. How are the coordinates $(x, y, z)$ and $\left(x^{\prime }, y^{\prime }, z^{\prime }\right)$ of an arbitrary point $M$ expressed in terms of each other, if the origin and basis vectors of the second system have the following coordinates in the first system $O x y z$: $O^{\prime }(-1,3,0), \mathbf{e}_{1}^{\prime } = (1,-1,0), \mathbf{e}_{2}^{\prime } = (2,2,-1), \mathbf{e}_{3}^{\prime } = (-1,1,1)$?

Two coordinate systems $O x y z$ and $O^{\prime } x^{\prime } y^{\prime } z^{\prime }$ are given in space. The coordinates $(x, y, z)$ and $\left(x^{\prime }, y^{\prime }, z^{\prime }\right)$ of an arbitrary point $M$ relative to these two systems are related by the following transition formulas:

In an affine coordinate system, the equations are given of three planes $A_{1} x+B_{1} y+C_{1} z+D_{1} = 0, A_{2} x+B_{2} y+C_{2} z+D_{2} = 0, A_{3} x+B_{3} y + C_{3} z+D_{3} = 0$, meeting at one point and not belonging to one bundle, and a point $E(x_{0}, y_{0}, z_{0})$ not lying on any of these planes. Taking these planes, respectively, as the coordinate planes $O^{\prime } y^{\prime } z^{\prime }, O^{\prime } x^{\prime } z^{\prime }, O^{\prime } x^{\prime } y^{\prime }$, find the formulas expressing the coordinates $(x^{\prime }, y^{\prime }, z^{\prime })$ of an arbitrary point $M$ in terms of its coordinates $(x, y, z)$, if the coordinates of the point $E$ in the system $O^{\prime } x^{\prime } y^{\prime } z^{\prime }$ equal $(1,1,1)$.

Two rectangular coordinate systems with a common origin are given in space: the axis $O^{\prime } x^{\prime }$ coincides with the axis $O z$, the axis $O^{\prime } y^{\prime }$ with the axis $O x$, but has the opposite direction to it, and the axis $O^{\prime } z^{\prime }$ coincides with the axis $O y$ and likewise has the opposite direction to it. Find the formulas expressing the coordinates $(x^{\prime }, y^{\prime }, z^{\prime })$ of an arbitrary point $M$ in terms of the coordinates $(x, y, z)$.

Relative to the original rectangular coordinate system in space, the directions of the coordinate axes of a new rectangular system are given by the vectors: the axis $O^{\prime } x^{\prime }$ is directed along the vector $(-2,-6,3)$, the axis $O^{\prime } y^{\prime }$ along $(6,-3,-2)$, and the axis $O^{\prime } z^{\prime }$ along $(-3,-2,-6)$. The origin $O^{\prime }$ of the new coordinate system is at the point $(-2,4,1)$. Find the formulas expressing the old coordinates $(x, y, z)$ of an arbitrary point $M$ in terms of its new coordinates $\left(x^{\prime }, y^{\prime }, z^{\prime }\right)$.

Two coordinate systems are given: $O x y z$ with angles between axes of $60^{\circ }$ and $O x^{\prime } y^{\prime } z^{\prime }$ with angles between axes of $90^{\circ }$, with a common origin $O$ and basis vectors of equal length. Write the formulas for passing from the oblique coordinate system $O x y z$ to the rectangular system $O x^{\prime } y^{\prime } z^{\prime }$, if the axes $O x$ and $O x^{\prime }$ of both systems coincide, the axis $O y^{\prime }$ lies in the plane $O x y$ and forms an angle $30^{\circ }$ with the axis $O y$, and the rays determining the positive directions of the axes $O z$ and $O z^{\prime }$ lie on the same side of the plane $O x y$.

Two coordinate systems $O x_{1} x_{2} x_{3}$ and $O x_{1}^{\prime } x_{2}^{\prime } x_{3}^{\prime }$ are given, with a common origin $O$ and basis vectors of equal length along all axes of both systems. The cosines of the angles between the axes of the first system equal, respectively, $\cos \left(x_{1} O x_{2}\right) = \omega_{12}, \cos \left(x_{2} O x_{3}\right) = \omega_{23}, \cos \left(x_{3} O x_{1}\right) = \omega_{31}$. The cosines of the angles between the axes of the first and second systems are given by the table

Write the formulas relating the coordinates $x_{1}, x_{2}, x_{3}$ and $x_{1}^{\prime }, x_{2}^{\prime }, x_{3}^{\prime }$ of one and the same point $M$ in both systems.

Find the formulas for passing from one rectangular coordinate system $O x y z$ to another $O^{\prime } x^{\prime } y^{\prime } z^{\prime }$, if the origin of the second system is at the point $O^{\prime }(1,2,3)$, the systems have opposite orientation, and the following holds:

Relative to a rectangular coordinate system, the equations of the coordinate planes of a new system are given:

Verify that these planes are pairwise perpendicular, and express the new rectangular coordinates of an arbitrary point $M$ in terms of its old coordinates, under the condition that the old origin $O$ has positive coordinates in the new system.

Relative to a rectangular coordinate system, the coordinate planes of a new system $O^{\prime } x^{\prime } y^{\prime } z^{\prime }$ are given:

Verify that these planes are pairwise perpendicular, and express the new rectangular coordinates of an arbitrary point $M$ in terms of its old coordinates, under the condition that the point having coordinates $(1,-1,-1)$ in the old system has positive coordinates in the new system.