Formulas and Tables

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:
    \(d = AB =\) \( \left| {{x_2} – {x_1}} \right| =\) \( \left| {{x_1} – {x_2}} \right|\)
The distance between two points on the coordinate line
  1. 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 \(B\).
Dividing a line segment in the ratio lambda
  1. In the special case when \(\lambda = 1\), the previous formula allows to calculate the coordinate of the midpoint of the segment:
    \({x_0} = {\large\frac{{{x_1} + {x_2}}}{2}\normalsize},\;\) \(\lambda = 1.\)