# Formulas

## Analytic Geometry # One-Dimensional Coordinate System

Point coordinates: $${x_0}$$, $${x_1}$$, $${x_2}$$
Distance between two points: $$d$$
Real number (length ratio): $$\lambda$$
1. The coordinates of a point are a set of numbers which determine the position of the point in a set (on a plane, in a space, or on a manifold). A system in which different points have unique coordinates is called a coordinate system or a system of coordinates.
2. In geometry, the most common is the Cartesian coordinate system.
It is defined by its origin and a basis (basis vectors). If the basis vectors (in an $$n$$-dimensional coordinate system) are mutually perpendicular to each other, such coordinate system is called a rectangular Cartesian system.
3. A one-dimensional coordinate system is defined by its origin and a single basis vector that defines the positive direction of the coordinate axis ($$x$$-axis). The coordinates of any point in such a system are determined by a single real number.
4. The distance between two points $$A\left( {{x_1}} \right)$$ and $$B\left( {{x_2}} \right)$$ on the coordinate line is equal to the absolute value of the difference of their coordinates:
$$d = AB =$$ $$\left| {{x_2} – {x_1}} \right| =$$ $$\left| {{x_1} – {x_2}} \right|$$
5. Dividing a line segment in the ratio $$\lambda$$
Let the point $$C\left( {{x_0}} \right)$$ divide the line segment $$AB$$ in the ratio $$\lambda$$. Then the coordinate $${x_0}$$ of the point $$C$$ is given by the formula
$${x_0} = {\large\frac{{{x_1} + \lambda {x_2}}}{{1 + \lambda }}\normalsize},\;$$ $$\lambda = {\large\frac{{AC}}{{CB}}\normalsize},\;$$ $$\lambda \ne – 1,$$
where $${x_1}$$ is the coordinate of the point $$A$$ and $${x_2}$$ is the coordinate of the point $$B$$.
6. In the special case when $$\lambda = 1$$, the previous formula allows to calculate the coordinate of the midpoint of the segment:
$${x_0} = {\large\frac{{{x_1} + {x_2}}}{2}\normalsize},\;$$ $$\lambda = 1.$$