Vectors and 3D Geometry Summary

Calculus 3 · Axiom Academy

SUMMARY Vectors and 3D Geometry Vectors give us the language for three-dimensional geometry — measuring alignment, building perpendicular directions, and describing lines, planes, and surfaces. Vectors unify geometry: position, direction, and normal vectors turn geometric questions into algebra. The dot product measures alignment: it finds angles, tests orthogonality, and projects one vector onto another. The cross product creates perpendicularity: it builds a vector normal to two others and measures the area they span. Parametric equations describe lines: a point plus a direction vector traces every line in space. Normal vectors define planes: one vector perpendicular to a plane fixes its orientation completely. Second-degree equations classify surfaces: the quadric family names the fundamental 3D shapes. Core Concept Vector Fundamentals A vector carries both magnitude and direction, written in components as . Its magnitude is its length; dividing by that length gives the unit vector pointing the same way. Operations: add component-wise; scalar multiplication scales length (and flips direction if negative). Position vectors: a vector from the origin to a point gives that point its coordinates. The dot product returns a scalar. It is positive when vectors point similar ways, negative when opposite, and exactly zero when they are perpendicular — the orthogonality test. When to use: to find an angle, test for perpendicularity, or project.

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