Coordinate Systems in the Plane and in Space

In a rectangular coordinate system $x, y$ in the plane, a point $M(x, y)$ is given. Find the point symmetric to $M$ with respect to:

In a rectangular coordinate system $x, y, z$ in space, a point $M(x, y, z)$ is given. Find the point symmetric to $M$ with respect to:

Two parallel lines $\ell_{1}$ and $\ell_{2}$ are given, the distance between which equals $d$, together with three points $A, B$ and $C$. Points $A$ and $B$ are symmetric with respect to the line $\ell_{1}$, and points $B$ and $C$ are symmetric with respect to $\ell_{2}$. Find the distance between the points $A$ and $C$.

Given the position vectors $\mathbf{r}_{1}, \mathbf{r}_{2}, \mathbf{r}_{3}$ of three consecutive vertices of a parallelogram, find the position vector $\mathbf{r}_{4}$ of the fourth vertex.

Given the position vectors $\mathbf{r}_{A}, \mathbf{r}_{B}, \mathbf{r}_{C}$ of three consecutive vertices of a trapezoid $A B C D$ and the ratio of the bases

find the position vector $\mathbf{r}_{D}$ of the fourth vertex.

A point $C$ is said to divide the segment $A B$ in the ratio $\lambda$ if the equality

holds. Given the position vectors $\mathbf{r}_{A}$ and $\mathbf{r}_{B}$ of the points $A$ and $B$, find:

Within what limits does the number $\lambda$, equal to the ratio in which a point $M$ divides the segment $A B$, lie if $M$ is located:

Given the position vectors $\mathbf{r}_{A}, \mathbf{r}_{B}, \mathbf{r}_{C}$ of the vertices of a triangle $A B C$, find the position vector $\mathbf{r}$ of the point where its medians meet.

In space, two points $A(1,2,3)$ and $B(7,2,5)$ are given. On the line $A B$ find a point $M$ such that the points $B$ and $M$ lie on opposite sides of the point $A$, and the segment $A M$ is twice as long as the segment $A B$.

In a triangle $A B C$, the bisector $A D$ of the angle $A$ is drawn. Express the position vector $\mathbf{r}_{D}$ of the point $D$ in terms of the position vectors $\mathbf{r}_{B}, \mathbf{r}_{C}$ of the points $B$ and $C$ and the side lengths $b = \left|A C\right|, c = \left|A B\right|$.

Given the coordinates $A(1,1), B(3,5), C(2,6)$ of three consecutive vertices of a trapezoid $A B C D$ and the ratio of the bases $\left|B C\right|:\left|A D\right| = 2$, find the coordinates of the vertex $D$, of the point $M$ where the diagonals meet, and of the point $S$ where the extensions of the lateral sides meet.

In a parallelogram $A B C D$, the midpoints $E, F$ of the sides $A B$ and $C D$, respectively, are marked. A line $\ell$, not passing through the point $E$, meets the lines $A B, E F, C E$ and $D E$ at the points $P, Q, R$ and $S$, respectively. The point $R$ divides the segment $P Q$ in the ratio $\lambda$. In what ratio does the point $S$ divide this segment?

Three points $A, B$ and $C$ are given. Let $\mathbf{r}_{A}, \mathbf{r}_{B}$ and $\mathbf{r}_{C}$ be their position vectors. Which of the following expressions define a vector that does not depend on the choice of origin, and which define the position vector of a point that does not depend on the choice of origin? Indicate these vectors and points:

Does the line through the points $A(6,-8,-1)$ and $B(-3,4,3)$ intersect any coordinate axis?

Prove that the segments joining the midpoints of opposite edges of a tetrahedron meet at one point and are bisected by that point; prove also that the segments joining the vertices of the tetrahedron to the points where the medians of the opposite faces meet pass through the same point, which divides them in the ratio $3:1$ (measured from the vertices).

Two points $A$ and $B$ are given, the distance between which equals $2 c$. Find the locus of points for which the sum of the squares of the distances to $A$ and $B$ equals $2 a^{2}$, under the condition $a > c$.

Two points $A$ and $B$ are given, the distance between which equals $2 c$. Find the locus of points for which the absolute value of the difference of the squares of the distances to $A$ and $B$ equals $4 a^{2}$.

Find the locus of points for which the sum of the squares of the distances to the vertices of the acute angles of an isosceles right triangle is twice the square of the distance to the vertex of the right angle.

Find the locus of points for which the sum of the squares of the distances to the three vertices of an equilateral triangle is constant, under the condition that this locus contains the midpoint of one of the sides of the triangle.

Find the locus of points for which the sum of the squares of the distances to two vertices $A$ and $B$ of a triangle $A B C$ equals the square of the distance to its third vertex $C$.

Find the locus of points for which the sum of the squares of the distances to the three vertices of a triangle $A B C$ equals $a^{2}$.

Prove that the locus of points for which the sum of the squares of the distances to several fixed points is constant is a circle or a single point.

Two distinct points $A$ and $B$ and a positive number $k \neq 1$ are given. Find the locus of points for which the ratio of the distances to $A$ and $B$ equals $k$.

Find the locus of points for which the sum of the distances to two opposite vertices of a rectangle equals the sum of the distances to the other two opposite vertices.

Two circles with centers $O_{1}, O_{2}$ and radii $r_{1}$ and $r_{2}$, respectively, are given. Find the locus of points from which equal tangents can be drawn to them.

A circle with center $O$ and radius $r$ is given, together with a point $A$ at a distance $a$ from $O$. Find the locus of points from which the tangents drawn to the given circle are equal to the segments joining these points to $A$.

Two circles $x^{2}+y^{2}-6 x-16 = 0, x^{2}+y^{2}+8 x-2 = 0$ are given. Find the locus of points from which equal tangents can be drawn to these circles.

Two circles $x^{2}+y^{2}-6 x-27 = 0, x^{2}+y^{2}+2 x-8 = 0$ are given. Find the locus of points from which the tangents drawn to the larger circle are twice as long as the tangents to the smaller circle.

Find the locus of points for which the square of the distance to the point of intersection of two given mutually perpendicular lines is $5/2$ times the product of the distances to these lines.

Find the locus of points for which the sum of the distances to the coordinate axes is constant, under the condition that this locus contains the point $(2,-1)$.

Find the locus of points for which the product of the distances to the lines containing two opposite sides of a square equals the product of the distances to the lines containing the other two sides.

Find the locus of points for which the sum of the distances to two given parallel lines is twice the distance to a third given line perpendicular to the first two.

A rectangle with sides $2 a$ and $2 b$ is given, where $a > b$. Find the locus of points for which the sum of the distances to the lines containing two opposite sides of the rectangle equals the sum of the distances to the lines containing the other two opposite sides.

A regular hexagon $A B C D E F$ with side length 1 is given. Taking the vertex $A$ as the pole, the direction of the vector $\overrightarrow {A B}$ as the positive direction of the polar axis, and the direction of the shortest rotation from $\overrightarrow {A B}$ to $\overrightarrow {A C}$ as the positive direction for measuring angles, find the polar coordinates of the vertices of the hexagon in this system.

Compute the distance between two points given by their polar coordinates:

The polar coordinates of the points $A\left(8,-\frac{2 \pi }{3}\right)$ and $B\left(6, \frac{\pi }{3}\right)$ are given. Compute the polar coordinates of the midpoint of the segment $A B$.

A point $A\left(5, \frac{2 \pi }{3}\right)$ is given with respect to a polar coordinate system. Find:

In a polar coordinate system find the point symmetric to the point $(\rho_{0}, \varphi_{0})$ with respect to the line containing the ray $\varphi = \varphi_{1}$.

With respect to a polar coordinate system, the points $A\left(2, \frac{\pi }{6}\right), B\left(3, \frac{4 \pi }{3}\right), C\left(1, \frac{3 \pi }{2}\right), D(5, \pi ), E(5,0)$ are given. What coordinates will these points have if the polar axis is rotated about the pole in the positive direction through an angle of $3 \pi / 4$?

Compute the area $S$ of a triangle, one of whose vertices is at the pole and the other two have polar coordinates $\left(4, \frac{\pi }{9}\right),\left(1, \frac{5 \pi }{18}\right)$.

With respect to a polar coordinate system, the points $A\left(2, \frac{\pi }{3}\right), B\left(\sqrt{2}, \frac{3 \pi }{4}\right), C\left(5, \frac{\pi }{2}\right), D\left(3,-\frac{\pi }{6}\right)$ are given. Find the coordinates of these points in the corresponding rectangular coordinate system.

Given the rectangular coordinates of the points $A(-1,1), B(0,2)$, $C(5,0), D(-8,-6)$, find their coordinates in the polar coordinate system corresponding to the given rectangular one.

Given the polar coordinates of a point: $\rho = 10, \varphi = \pi / 6$, find its rectangular coordinates, if the pole of the polar coordinate system is at the point $(2,3)$ and the polar axis is parallel to the axis $O x$.

The pole of a polar coordinate system is at the point $(3,5)$, and the positive direction of the polar axis coincides with the positive direction of the axis $O y$. Find, in this system, the polar coordinates of the points $M_{1}(9,-1)$ and $M_{2}(5,5-2 \sqrt{3})$.

Write the expression for the rectangular coordinates in terms of the polar coordinates, if the pole of the polar system is at the point $(x_{0}, y_{0})$ and its axis is directed at an angle $\varphi_{0}$ to the axis $O x$.

Using a transition to a polar coordinate system, prove that under a rotation in the positive direction through an angle $\varphi$, the vector with coordinates $(x, y)$ becomes the vector with coordinates $(x \cos \varphi -y \sin \varphi , x \sin \varphi +y \cos \varphi )$.

One vertex of an equilateral triangle is at the point $(0,0)$, and the center is at the point $(2,4)$. Find the coordinates of the other vertices.

Two opposite vertices of a square, $(1,1)$ and $(3,5)$, are given. Find the other two vertices.

The coordinates of two adjacent vertices of a square $A B C D$ are given: $A(1,-1)$ and $B(3,1)$. Find the coordinates of the remaining vertices.

Find the equation of the line obtained from the line $x+3 y = 2$ by a rotation in the positive direction through an angle $\pi / 4$ about the point where it meets the abscissa axis.

The vertices $A$ and $B$ of a triangle $A B C$ have coordinates $(0,0)$ and $(28,21)$, respectively. Find the coordinates of the vertex $C$, knowing the angles: $\angle A = \frac{\pi }{4}$ and $\angle B = \arccos \frac{4}{5}$.

The vertex $A$ of an equilateral triangle $A B C$ is at the point $(0,1)$. Find the vertices $B$ and $C$, knowing that they lie on the lines $y = 3$ and $y = 4$, respectively.

Prove that in a polar coordinate system the equation

defines a line for any $\varphi_{0}$ and $a$. What is the geometric meaning of the quantities $\varphi_{0}, a$?

In a polar coordinate system a line is given by the equation

Find the equation of this line in rectangular coordinates. Find the polar coordinates of the point symmetric to the pole with respect to this line.

Set up the equation of the line $A B$ in a polar coordinate system, if the points $A$ and $B$ have polar coordinates:

The center of an equilateral triangle is at the origin, and one of its sides is given in a polar coordinate system by the equation $\rho = a / \cos \varphi$. Find the equations of the remaining sides of the triangle.

In polar coordinates, write the equation of a circle of radius $r_{0}$ centered at the point $(\rho_{0}, \varphi_{0})$.

Find the spherical coordinates of the points, given their rectangular coordinates:

Find the spherical coordinates of a point $M$, knowing that the ray $O M$ forms angles of $\pi / 4$ and $\pi / 3$ with the axes $O x$ and $O y$, respectively, and that the coordinate $z$ of the point $M$ equals $-1$.

Find the rectangular coordinates of a point lying on a sphere of radius 1, knowing its longitude $\varphi = 330^{\circ }$ and latitude $\theta = 45^{\circ }$.

Find the length of the shorter of the two great-circle arcs joining two points $A$ and $B$ lying on a sphere of radius $\rho$, knowing the longitude and latitude of these points: $A\left(\varphi_{1}, \theta_{1}\right), B\left(\varphi_{2}, \theta_{2}\right)$.

Find the cylindrical coordinates of the points, given their rectangular coordinates: $A(3,-4,5), B(1,-1,-1), C(-6,0,8)$.

Find the cylindrical coordinates of a point $M$, knowing that the ray $O M$ makes angles $\pi / 3, \pi / 3$ with the coordinate axes $O x$ and $O y$, that its angle with the axis $O z$ is acute, and that the length of the segment $O M$ equals 1. Find the angle between the ray $O M$ and the axis $O z$.

Find the angle $\alpha$ between the vector $\overrightarrow {O M}$ and the axis $O x$, knowing the cylindrical coordinates $\rho , \varphi , z$ of the point $M$.

In a triangle $A B C$ the medians $A D, B E$ and $C F$ are drawn. Find the sum of the vectors $\overrightarrow {A D}+\overrightarrow {B E}+\overrightarrow {C F}$.

Points $E$ and $F$ are the midpoints of the sides $A B$ and $C D$ of a quadrilateral $A B C D$. Prove that $\overrightarrow {E F} = (\overrightarrow {B C}+\overrightarrow {A D}) / 2$. Deduce from this the converse theorem on the midline of a trapezoid.

The vectors $\overrightarrow {A C} = \mathbf{a}$ and $\overrightarrow {B D} = \mathbf{b}$ are the diagonals of a parallelogram $A B C D$. Express in terms of the vectors $\mathbf{a}$ and $\mathbf{b}$ the vectors $\overrightarrow {A B}, \overrightarrow {B C}, \overrightarrow {C D}$ and $\overrightarrow {D A}$, which are the sides of this parallelogram.

Points $K$ and $L$ are the midpoints of the sides $B C$ and $C D$ of a parallelogram $A B C D$. Setting $\overrightarrow {A K} = \mathbf{k}$ and $\overrightarrow {A L} = \mathbf{l}$, express in terms of the vectors $\mathbf{k}$ and $\mathbf{l}$ the vectors $\overrightarrow {B C}$ and $\overrightarrow {C D}$.

The vectors $\overrightarrow {A B} = \mathbf{p}$ and $\overrightarrow {A F} = \mathbf{q}$ are two adjacent sides of a regular hexagon $A B C D E F$. Express in terms of $\mathbf{p}$ and $\mathbf{q}$ the vectors $\overrightarrow {B C}, \overrightarrow {C D}, \overrightarrow {D E}, \overrightarrow {E F}$ running along the sides of this hexagon.

In a triangle, find a point such that the sum of the vectors going from this point to the vertices of the triangle equals $\mathbf{0}$.

Two vectors $\overrightarrow {O A} = \mathbf{a}, \overrightarrow {O B} = \mathbf{b}$ issue from a point $O$. Find some vector $\overrightarrow {O M}$ lying along the bisector of the angle $A O B$. Deduce from this the angle bisector theorem.

A parallelepiped $A B C D A^{\prime } B^{\prime } C^{\prime } D^{\prime }$ is built on three noncoplanar vectors $\overrightarrow {A B} = \mathbf{p}, \overrightarrow {A D} = \mathbf{q}, \overrightarrow {A A^{\prime }} = \mathbf{r}$. Express in terms of $\mathbf{p}$, $\mathbf{q}$ and $\mathbf{r}$ the vectors coinciding with the edges, the diagonal of the parallelepiped, and the diagonals of the faces of this parallelepiped, for which the vertex $A^{\prime }$ serves as the initial point.

In a tetrahedron $A B C D$ the vectors of the edges issuing from the vertex $A$ are given: $\overrightarrow {A B} = \mathbf{b}, \overrightarrow {A C} = \mathbf{c}, \overrightarrow {A D} = \mathbf{d}$. Express in terms of them the vectors of the remaining edges of the tetrahedron, the median $D M$ of the face $B C D$, and the vector $\overrightarrow {A Q}$, where $Q$ is the point where the medians of the face $B C D$ meet.

A tetrahedron $O A B C$ is given. Setting $\overrightarrow {O A} = \mathbf{a}, \overrightarrow {O B} = \mathbf{b}, \overrightarrow {O C} = \mathbf{c}$, express in terms of $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ the vectors $\overrightarrow {M N}, \overrightarrow {P Q}$ and $\overrightarrow {R S}$, where $M, P$ and $R$ are the midpoints of the edges $O A, O B$ and $O C$, and $N, Q$ and $S$ are the midpoints of the corresponding opposite edges.

A skew quadrilateral $A B C D$ is given. The vectors $\overrightarrow {A B} = \mathbf{m}$ and $\overrightarrow {C D} = \mathbf{p}$ are known. Find the vector $\overrightarrow {E F}$ joining the midpoints of the diagonals $A C$ and $B D$.

Prove that the sum of the vectors going from the center of a regular polygon to its vertices equals zero.

Prove that the vector going from an arbitrary point of the plane to the center of a regular polygon is the arithmetic mean of the vectors going from this point to the vertices of the polygon.

Given the position vectors $\mathbf{r}_{A}, \mathbf{r}_{B}, \mathbf{r}_{D}$ and $\mathbf{r}_{A^{\prime }}$ of four vertices of a parallelepiped $A B C D A^{\prime } B^{\prime } C^{\prime } D^{\prime }$, find the position vectors of its four remaining vertices.

The position vectors $\overrightarrow {O A} = \mathbf{r}_{1}, \overrightarrow {O B} = \mathbf{r}_{2}$ and $\overrightarrow {O C} = \mathbf{r}_{3}$ are edges of a parallelepiped. Find the position vector of the point where the diagonal of the parallelepiped issuing from the vertex $O$ meets the plane passing through the vertices $A, B$ and $C$.

Three vectors $\mathbf{a} = (5,7,2), \mathbf{b} = (3,0,4), \mathbf{c} = (-6,1,-1)$ are given. Find the vectors:

Represent the vector $\mathbf{d}$ as a linear combination of the vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ in each of the following cases:

Determine in which of the following cases the triples of vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are linearly dependent, and, in the cases where this is possible, represent the vector $\mathbf{c}$ as a linear combination of the vectors $\mathbf{a}$ and $\mathbf{b}$:

Show that whatever the three vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ and the three numbers $\lambda , \mu , \nu$ may be, the vectors $\lambda \mathbf{a}-\mu \mathbf{b}, \nu \mathbf{b}-\lambda \mathbf{c}, \mu \mathbf{c}-\nu \mathbf{a}$ are coplanar.

Find the projection onto the plane $O y z$ of the vector $(1,3,5)$ parallel to the vector $(4,-1,3)$.

Find the projection of the vector $(7,5,-3)$ onto the plane determined by the vectors $(1,1,1)$ and $(2,-1,1)$, parallel to the vector $(2,-2,1)$.

Four vectors are given

$\mathbf{a} = (1,5,3), \quad \mathbf{b} = (6,-4,-2), \quad \mathbf{c} = (0,-5,7), \quad \mathbf{d} = (-20,27,-35)$.

Choose numbers $\alpha , \beta$ and $\gamma$ so that the vectors $\alpha \mathbf{a}, \beta \mathbf{b}, \gamma \mathbf{c}$ and $\mathbf{d}$ form a closed broken line, if the start of each subsequent vector is placed at the end of the preceding one.

Find the coordinates of the vertices of a parallelogram $A B C D$ in an affine coordinate system whose origin is at the center $O$ of this parallelogram, and whose basis vectors are $\overrightarrow {A B}, \overrightarrow {B O}$.

Find the affine coordinates of the vertices of a regular hexagon $A B C D E F$, taking the point $A$ as the origin and the pair of vectors $\overrightarrow {A B}, \overrightarrow {B C}$ as the basis.

In a trapezoid $A B C D$ the ratio of the lengths of the bases $A D$ and $B C$ equals $k$. Taking the vertex $A$ as the origin of coordinates, and the vectors $\overrightarrow {A D}$ and $\overrightarrow {A B}$ as the basis vectors, find the coordinates of the vertices of the trapezoid, of the point $M$ where the diagonals meet, and of the point $S$ where the extensions of the lateral sides meet.

Find the coordinates of the vertices of a tetrahedron $O A B C$ in a coordinate system with origin at the vertex $O$, whose basis vectors are the medians $\overrightarrow {O D}, \overrightarrow {O E}, \overrightarrow {O F}$ of the faces $B O C, C O A, A O B$.

Find the coordinates of the vertices of a tetrahedron $A B C D$, taking the point $P$ where the medians of the face $B C D$ meet as the origin, and the vectors $B Q, C R$, $D S$ as the basis vectors, where $Q, R$ and $S$ are the points where the medians of the faces $A C D, A B D$ and $A B C$ meet, respectively.

Find the coordinates of the center of gravity of a homogeneous rod $A O B$ bent at a right angle, if the lengths $\left|O A\right| = a$ and $\left|O B\right| = b$. Take the point $O$ as the origin of coordinates, and the vectors $\overrightarrow {O A}$ and $\overrightarrow {O B}$ as the unit vectors of the axes $O x$ and $O y$, respectively.

Find the center of gravity of a triangular wire frame.

Find the scalar product of the vectors $\mathbf{a}$ and $\mathbf{b}$ in each of the following cases:

In a triangle $A B C$ the side lengths $\left|B C\right| = 5,\left|C A\right| = 6$, $\left|A B\right| = 7$ are given. Find the scalar product of the vectors $\overrightarrow {A B}$ and $\overrightarrow {B C}$.

Using only the bilinearity and symmetry properties of the scalar product, find the angle $\alpha$ at the apex of an isosceles triangle, knowing that the medians drawn from the ends of the base of this triangle are mutually perpendicular.

Prove that the vectors $\mathbf{p} = \mathbf{a}(\mathbf{b}, \mathbf{c})-\mathbf{b}(\mathbf{a}, \mathbf{c})$ and $\mathbf{c}$ are perpendicular to each other.

What angle do the unit vectors $\mathbf{s}$ and $\mathbf{t}$ form, if it is known that the vectors $\mathbf{p} = \mathbf{s}+2 \mathbf{t}$ and $\mathbf{q} = 5 \mathbf{s}-4 \mathbf{t}$ are mutually perpendicular?

An equilateral triangle $A B C$ is given, whose side lengths equal 1. Setting $\overrightarrow {B C} = \mathbf{a}, \overrightarrow {C A} = \mathbf{b}, \overrightarrow {A B} = \mathbf{c}$, compute the value of the expression $(\mathbf{a}, \mathbf{b})+(\mathbf{b}, \mathbf{c})+(\mathbf{c}, \mathbf{a})$ using the properties of the scalar product.

In a triangle $A B C$ the medians $A D, B E$ and $C F$ are drawn. Compute $(\overrightarrow {B C}, \overrightarrow {A D})+(\overrightarrow {C A}, \overrightarrow {B E})+(\overrightarrow {A B}, \overrightarrow {C F})$.

In a right triangle $A B C$ a perpendicular $C H$ is dropped onto the hypotenuse $A B$. Express the vector $\overrightarrow {C H}$ in terms of the vectors $\mathbf{a} = \overrightarrow {C B}$ and $\mathbf{b} = \overrightarrow {C A}$.

A rectangle $A B C D$ and a point $M$ (which may lie either in the plane of the rectangle or outside it) are given. Show that:

In a triangle $A B C$ a point $D$ divides the side $A B$ in the ratio $\overrightarrow {A D}: \overrightarrow {D B} = \lambda$. Express the length of the segment $C D$ in terms of the side lengths of the triangle and the number $\lambda$.

Prove that for any arrangement of the points $A, B, C, D$ in the plane or in space, the equality

holds.

Prove that in a regular tetrahedron the segment joining the midpoints of opposite edges is their common perpendicular.

Prove that if, in a tetrahedron $A B C D$, the edge $A B$ is perpendicular to $C D$, and $A C$ is perpendicular to $B D$, then the edges $A D$ and $B C$ are also perpendicular.

Compute the length of the diagonal $O D$ of a parallelepiped, knowing the lengths $a = \left|O A\right|, b = \left|O B\right|, c = \left|O C\right|$ of three edges issuing from the same vertex, and the plane angles $\alpha = \angle B O C, \beta = \angle A O C, \gamma = \angle A O B$ between them.

Compute the scalar product of the vectors $\mathbf{a}$ and $\mathbf{b}$, given by their coordinates, in each of the following cases: