Lines and Vectors in 2D: Equations, Slopes, and Relationships

Fundamentals of 2D Analytical Geometry

This document outlines key concepts in the analytical geometry of straight lines within a two-dimensional plane.

Coordinate Systems and Vector Representation

An affine coordinate system (SdR) is defined by a triple (O, u, U), where O is a fixed point in the Euclidean plane (E2) serving as the origin, and (u, U) forms a basis for the vector space R² (V2R). A point P in E2 is uniquely determined by its coordinates (x, y) relative to the chosen basis, which represent its position vector OP.

Points and Segments

The coordinates of the midpoint M of a segment AB are obtained by averaging the coordinates of its endpoints. If A = (x_A, y_A) and B = (x_B, y_B), then the midpoint M = ((x_A + x_B) / 2, (y_A + y_B) / 2).

Equations of a Straight Line

A straight line in R² can be represented by various equations, often defined by a known point on the line and a direction vector.

• Vector Equation

  Given a point x₁ on the line and a direction vector v, any point x on the line can be expressed as: x = x₁ + tv, where t is a scalar parameter.

• Parametric Equations

  If x₁ = (x₁, y₁) and v = (v₁, v₂), the vector equation expands into parametric equations:

  • x = x₁ + tv₁
  • y = y₁ + tv₂

• Continuous Equation

  By eliminating the parameter t from the parametric equations, we obtain the continuous equation:

  (x - x₁) / v₁ = (y - y₁) / v₂

  This form is valid when v₁ ≠ 0 and v₂ ≠ 0.

• General Equation

  The general (or implicit) equation of a line is given by: ax + by + c = 0. From this form, a direction vector can be derived as (-b, a) or (b, -a).

• Point-Slope Form

  If a line passes through a point (x₁, y₁) and has a slope m, its equation is: y - y₁ = m(x - x₁). The slope m is equivalent to v₂/v₁ from the direction vector.

Line Properties: Inclination and Slope

The inclination of a line is the angle it forms with the positive x-axis, measured counter-clockwise, typically between 0° and 180°. The slope (gradient) of a line, denoted by m, is the tangent of its inclination angle (m = tan θ).

If P₁(x₁, y₁) and P₂(x₂, y₂) are two distinct points on a line, the slope is calculated as: m = (y₂ - y₁) / (x₂ - x₁). The slope is independent of the specific points chosen on the line. For three points P₁, P₂, P₃ to be collinear, the slopes between any two pairs must be equal.

Explicit Equation: Slope-Intercept Form

The slope-intercept form of a line's equation is: y = mx + b. Here, m is the slope, and b is the y-intercept, which is the y-coordinate where the line crosses the Y-axis (at point (0, b)).

Special Cases of Lines

• Lines Parallel to Axes

  • Parallel to X-axis: A horizontal line has a slope m = 0. Its equation is y = k, where k is a constant.
  • Parallel to Y-axis: A vertical line has an undefined slope. Its equation is x = k, where k is a constant.

• Angle Bisectors

  • First Bisector: Passes through the origin, has an inclination of 45°, and a slope m = 1. Equation: y = x.
  • Second Bisector: Passes through the origin, has an inclination of 135°, and a slope m = -1. Equation: y = -x.

Segment (Canonical) Equation

If a line intersects the X-axis at A(a, 0) and the Y-axis at B(0, b), its segment (or canonical) equation is: x/a + y/b = 1. This form is particularly useful when the x and y-intercepts are known.

Equation of a Line Through Two Points

Given two points P₁(x₁, y₁) and P₂(x₂, y₂), the equation of the line passing through them can be found using the point-slope form. First, calculate the slope m = (y₂ - y₁) / (x₂ - x₁). Then, use either point (e.g., P₁) in the point-slope form: y - y₁ = m(x - x₁). This method is not applicable for vertical lines (where x₁ = x₂), which have the form x = k.

Relationships Between Lines

For two lines given by their general equations: L₁: a₁x + b₁y + c₁ = 0 and L₂: a₂x + b₂y + c₂ = 0:

• Parallel Lines

  Two lines are parallel if their slopes are equal (m₁ = m₂) or, in general form, if a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

• Coincident Lines

  Two lines are coincident (the same line) if a₁/a₂ = b₁/b₂ = c₁/c₂.

• Intersecting Lines

  Two lines intersect at a single point if their slopes are different (m₁ ≠ m₂) or, in general form, if a₁/a₂ ≠ b₁/b₂.

• Perpendicular Lines

  Two lines are perpendicular if the product of their slopes is -1 (m₁ * m₂ = -1) or, in general form, if a₁a₂ + b₁b₂ = 0.