Vector geometry

Vector geometry describes geometric relationships using vectors and operations such as addition, scalar multiplication, and the dot product. It represents points, directions, distances, and angles algebraically.

Vector: A quantity with magnitude and direction, represented by components or as a directed displacement. Vectors are the basic objects used to represent directions and displacements in geometry.
Vector addition: The operation that combines vectors by adding their corresponding components. Adding displacements constructs longer paths and parallelograms.
Distance from a point to a line: The shortest Euclidean distance between a point and a line. Vector projection gives a direct calculation of the perpendicular separation.
Synthetic geometry: A method of proving geometric results through axioms, constructions, and logical relationships rather than coordinates. It reaches geometric conclusions without translating figures into vector equations.
Euclidean space: A space in which points have distances and angles governed by Euclidean geometry. It supplies the setting where vector lengths, angles, and positions have familiar geometric meanings.
Scalar multiplication: The operation that multiplies every component of a vector by the same scalar. Scaling changes a vector's length and can reverse its direction.
Line-plane intersection: The point or set of points shared by a line and a plane. Substituting a line's vector parameterization into a plane equation locates the intersection.
Coordinate geometry: The study of geometry using coordinate systems and algebraic equations. Vector geometry is a coordinate-based approach that emphasizes vector operations and relations.
Coordinate system: A scheme that assigns numerical coordinates to points in a space. Coordinates turn geometric points and vectors into numerical data for calculation.
Parametric equation: An equation that describes quantities as functions of one or more parameters. A parameterized vector expression traces lines and other geometric sets.

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