With a reference system made up of two perpendicular lines that intersect at the origin, each point on the plane can be "named" by two numbers: (x, y), which are the coordinates of the point, called abscissa and ordinate, respectively, which are the orthogonal distances of said point with respect to the Cartesian axes.

The equation of the axis is , and that of the axis is , lines that intersect at the origin, whose coordinates are .

The axis is also called abscissa axis, and the axis is ordinate axis. The axes divide space into four quadrants I, II, III and IV, in which the signs of the coordinates alternate from positive to negative (for example, the two coordinates of point A will be positive, while those of point B will both be negative).

The coordinates of any point will be given by the projections of the segment between the origin and the point on each of the axes.

On each of the axes unit vectors (i and j) are defined as those parallel to the axes and of module "Module (vector)") (length) the unit. In vector form, the position of point A is defined with respect to the origin with the components of the vector OA.

The position of point A will be:.

Note that the list of coordinates can express both the position of a point and the components of a vector in matrix notation "Matrix (mathematical)").

The distance between any two points will be given by the expression:.

Application of the Pythagorean theorem to the right triangle ABC.

Any vector AB will be defined by subtracting, coordinate by coordinate, those of the point of origin from those of the destination point:.

Obviously, the module of the vector AB will be the distance d between points A and B calculated above.

If we have a reference system formed by three mutually perpendicular lines (X, Y, Z) (abscissa, ordinate and coordinate), which intersect at the origin (0, 0, 0), each point in space can be named by three numbers: (x, y, z), called coordinates of the point, which are the orthogonal distances to the three main planes: those containing the pairs of axes YZ, XZ and YX, respectively.

The XY reference planes (z = 0); XZ (y = 0); and YZ (x = 0) divide the space into eight octants "Octant (geometry)") in which, as in the previous case, the signs of the coordinates can be positive or negative.

The generalization of the previous relations to the spatial case is immediate considering that a third coordinate (z) is now necessary to define the position of the point.

The coordinates of point A will be:.

and the B:.

The distance between points A and B will be:.

The segment AB will be:.

Contenido

Tanto en el caso plano como en el caso espacial pueden considerarse tres transformaciones elementales: traslación "Traslación (geometría)") del origen, rotación alrededor de un eje y escalado.

Origin translation

Assuming an initial coordinate system S1 with origin at O and axes x and y.

and the coordinates of a given point A, are in the system S1:.

given a second reference system S2.

The coordinate centers of the systems 0 and 0´ being different points, and the axes x, x´; e y, y´ parallel two by two, and the coordinates of O´, with respect to S1:.

It is called translation of the origin, to calculate the coordinates of A in S2, according to the previous data, which we will call:.

Given the points O, O´ and A, we have the sum of vectors:.

clearing

Which is the same as:

Separating the vectors by coordinates:.

and expanding it to three dimensions:.

Rotation around the origin

Given a coordinate system in the S plane with origin at O and x and y axes:.

and an orthonormal basis of this system:.

A point A on the plane will be represented in this system according to its coordinates:.

For a second reference system S rotated by an angle with respect to the first:.

and with an orthonormal basis:.

The calculation of the coordinates of point A, with respect to this second reference system, rotated with respect to the first, is called rotation around the origin, its representation being:.

It must be taken into account that the point and are the same point, ; One denomination or another is used to indicate the reference system used. The value of the coordinates with respect to one or another system is different, and that is what is intended to be calculated.

The representation of B in B is:.

Since point A in B1 is:.

With the previous transformation we have:

And, undoing the parentheses:

reordering:.

As:.

We have to:

As we knew:

By identification of terms:

Which are the coordinates of A in B, based on the coordinates of A in B and of .

Scaling

Let be a point with coordinates (x,y) in the plane. If both axes are scaled by a factor λ, the coordinates of that point in the new coordinate system will become:.

The scale factor λ does not necessarily have to be the same for both axes.

Being [T] the matrix "Matrix (mathematical)") of transformation and whose rows are also the components of the unit vectors i ' and j ' with respect to the original i and j, or if preferred, whose columns are the components of the original unit vectors in the rotated reference system.

Note: Vector quantities are in bold.

They are called Cartesian coordinates in honor of René Descartes (1596-1650), the famous French philosopher and mathematician who wanted to base his philosophical thinking on the method of taking an obvious "starting point" on which he would build all knowledge.

As the creator of analytical geometry, Descartes also began by taking a "starting point" in this discipline, the Cartesian reference system, in order to represent plane geometry, which uses only two mutually perpendicular lines that intersect at a point called the "origin of coordinates."

Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.[1]​.

The development of the Cartesian coordinate system would play a fundamental role in the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz.[2]​ The description of the plane in two coordinates was later generalized in the concept of vector spaces.[3]​.

Since Descartes many other coordinate systems have been developed, such as the polar coordinates for the plane, and the spherical and the cylindrical coordinates for three-dimensional space.

With a reference system made up of two perpendicular lines that intersect at the origin, each point on the plane can be "named" by two numbers: (x, y), which are the coordinates of the point, called abscissa and ordinate, respectively, which are the orthogonal distances of said point with respect to the Cartesian axes.

The equation of the axis is , and that of the axis is , lines that intersect at the origin, whose coordinates are .

The axis is also called abscissa axis, and the axis is ordinate axis. The axes divide space into four quadrants I, II, III and IV, in which the signs of the coordinates alternate from positive to negative (for example, the two coordinates of point A will be positive, while those of point B will both be negative).

The coordinates of any point will be given by the projections of the segment between the origin and the point on each of the axes.

On each of the axes unit vectors (i and j) are defined as those parallel to the axes and of module "Module (vector)") (length) the unit. In vector form, the position of point A is defined with respect to the origin with the components of the vector OA.

The position of point A will be:.

Note that the list of coordinates can express both the position of a point and the components of a vector in matrix notation "Matrix (mathematical)").

The distance between any two points will be given by the expression:.

Application of the Pythagorean theorem to the right triangle ABC.

Any vector AB will be defined by subtracting, coordinate by coordinate, those of the point of origin from those of the destination point:.

Obviously, the module of the vector AB will be the distance d between points A and B calculated above.

If we have a reference system formed by three mutually perpendicular lines (X, Y, Z) (abscissa, ordinate and coordinate), which intersect at the origin (0, 0, 0), each point in space can be named by three numbers: (x, y, z), called coordinates of the point, which are the orthogonal distances to the three main planes: those containing the pairs of axes YZ, XZ and YX, respectively.

The XY reference planes (z = 0); XZ (y = 0); and YZ (x = 0) divide the space into eight octants "Octant (geometry)") in which, as in the previous case, the signs of the coordinates can be positive or negative.

The generalization of the previous relations to the spatial case is immediate considering that a third coordinate (z) is now necessary to define the position of the point.

The coordinates of point A will be:.

and the B:.

The distance between points A and B will be:.

The segment AB will be:.

Contenido

Tanto en el caso plano como en el caso espacial pueden considerarse tres transformaciones elementales: traslación "Traslación (geometría)") del origen, rotación alrededor de un eje y escalado.

Origin translation

Assuming an initial coordinate system S1 with origin at O and axes x and y.

and the coordinates of a given point A, are in the system S1:.

given a second reference system S2.

The coordinate centers of the systems 0 and 0´ being different points, and the axes x, x´; e y, y´ parallel two by two, and the coordinates of O´, with respect to S1:.

It is called translation of the origin, to calculate the coordinates of A in S2, according to the previous data, which we will call:.

Given the points O, O´ and A, we have the sum of vectors:.

clearing

Which is the same as:

Separating the vectors by coordinates:.

and expanding it to three dimensions:.

Rotation around the origin

Given a coordinate system in the S plane with origin at O and x and y axes:.

and an orthonormal basis of this system:.

A point A on the plane will be represented in this system according to its coordinates:.

For a second reference system S rotated by an angle with respect to the first:.

and with an orthonormal basis:.

The calculation of the coordinates of point A, with respect to this second reference system, rotated with respect to the first, is called rotation around the origin, its representation being:.

It must be taken into account that the point and are the same point, ; One denomination or another is used to indicate the reference system used. The value of the coordinates with respect to one or another system is different, and that is what is intended to be calculated.

The representation of B in B is:.

Since point A in B1 is:.

With the previous transformation we have:

And, undoing the parentheses:

reordering:.

As:.

We have to:

As we knew:

By identification of terms:

Which are the coordinates of A in B, based on the coordinates of A in B and of .

Scaling

Let be a point with coordinates (x,y) in the plane. If both axes are scaled by a factor λ, the coordinates of that point in the new coordinate system will become:.

The scale factor λ does not necessarily have to be the same for both axes.

Being [T] the matrix "Matrix (mathematical)") of transformation and whose rows are also the components of the unit vectors i ' and j ' with respect to the original i and j, or if preferred, whose columns are the components of the original unit vectors in the rotated reference system.

Note: Vector quantities are in bold.