Page 1: Vector and Parametric Equations of Lines

This page introduces fundamental concepts of vector and parametric equations of lines in coordinate geometry. The content focuses on different representations of lines using vectors and parameters.

Definition: A vector equation of a line represents all points on the line using a point P and a direction vector v, written as (x,y) = P + k·v, where k is a real number parameter.

Example: For a line passing through point P(2,3) with direction vector (4,5), the vector equation is (x,y) = (2,3) + k(4,5).

Vocabulary: Parametric equations are obtained by separating the x and y coordinates from the vector equation, resulting in two separate equations: x = x₁ + k·vx and y = y₁ + k·vy.

Highlight: The parameter k can take any real number value, generating all points on the line through point P in the direction of vector v.