# 3D Coordinates & Vectors

Canonical: https://manyspace.aporia.systems/topics/3d-coordinates-vectors-mth-multc-168
Updated: 2026-10-13

Mathematics · Multivariable Calculus · MTH-MULTC-168

- Difficulty: 3 / 5
- Importance: 5 / 5
- Study time: 3–6 h

dot product, cross product, planes and lines in space

## Why it matters

Extends calculus to functions of several variables; essential for electromagnetism, fluid mechanics, GR, optimization and ML.

## Where it is used

Electromagnetic fields, fluid flow, gradient descent in ML, economic surfaces, thermodynamics potentials, graphics rendering.

## How you know you have learned it

Complete 3 independent examples correctly + explain common failure modes.

## What you need first

- [Integration by Parts](https://manyspace.aporia.systems/topics/integration-by-parts-mth-calc2-140) (required)
- [Gaussian Elimination](https://manyspace.aporia.systems/topics/gaussian-elimination-mth-linalg-155) (helpful)

## What it opens

- [Linear Programs](https://manyspace.aporia.systems/topics/linear-programs-mth-opt-234)
- [Simplex Method](https://manyspace.aporia.systems/topics/simplex-method-mth-opt-235)
- [Duality](https://manyspace.aporia.systems/topics/duality-mth-opt-236)
- [Convex Sets & Functions](https://manyspace.aporia.systems/topics/convex-sets-functions-mth-opt-237)
- [Convex Optimization Problems](https://manyspace.aporia.systems/topics/convex-optimization-problems-mth-opt-238)
- [KKT Conditions](https://manyspace.aporia.systems/topics/kkt-conditions-mth-opt-239)
- [Optimization Algorithms](https://manyspace.aporia.systems/topics/optimization-algorithms-mth-opt-240)
- [Least Squares & Regression](https://manyspace.aporia.systems/topics/least-squares-regression-mth-opt-241)
- [Combinatorial Optimization Intro](https://manyspace.aporia.systems/topics/combinatorial-optimization-intro-mth-opt-242)
- [Duality in Economics](https://manyspace.aporia.systems/topics/duality-in-economics-mth-opt-243)
- [Classification of PDEs](https://manyspace.aporia.systems/topics/classification-of-pdes-mth-pde-296)
- [First-Order PDEs](https://manyspace.aporia.systems/topics/first-order-pdes-mth-pde-297)
- [The Wave Equation](https://manyspace.aporia.systems/topics/the-wave-equation-mth-pde-298)
- [The Heat Equation](https://manyspace.aporia.systems/topics/the-heat-equation-mth-pde-299)
- [Laplace & Poisson Equations](https://manyspace.aporia.systems/topics/laplace-poisson-equations-mth-pde-300)
- [Separation of Variables](https://manyspace.aporia.systems/topics/separation-of-variables-mth-pde-301)
- [Fourier Series & Transforms](https://manyspace.aporia.systems/topics/fourier-series-transforms-mth-pde-302)
- [Green's Functions](https://manyspace.aporia.systems/topics/green-s-functions-mth-pde-303)
- [Sturm-Liouville Theory](https://manyspace.aporia.systems/topics/sturm-liouville-theory-mth-pde-304)
- [Introduction to Nonlinear PDEs](https://manyspace.aporia.systems/topics/introduction-to-nonlinear-pdes-mth-pde-305)
- [Parametrized Curves](https://manyspace.aporia.systems/topics/parametrized-curves-mth-diffg-321)
- [Surfaces in R3](https://manyspace.aporia.systems/topics/surfaces-in-r3-mth-diffg-322)
- [Second Fundamental Form & Curvature](https://manyspace.aporia.systems/topics/second-fundamental-form-curvature-mth-diffg-323)
- [Geodesics](https://manyspace.aporia.systems/topics/geodesics-mth-diffg-324)
- [Smooth Manifolds](https://manyspace.aporia.systems/topics/smooth-manifolds-mth-diffg-325)
- [Differential Forms](https://manyspace.aporia.systems/topics/differential-forms-mth-diffg-326)
- [Riemannian Metrics](https://manyspace.aporia.systems/topics/riemannian-metrics-mth-diffg-327)
- [Lie Groups & Lie Algebras Intro](https://manyspace.aporia.systems/topics/lie-groups-lie-algebras-intro-mth-diffg-328)
- [One-Dimensional Systems](https://manyspace.aporia.systems/topics/one-dimensional-systems-mth-dyn-329)
- [Discrete Maps & Chaos](https://manyspace.aporia.systems/topics/discrete-maps-chaos-mth-dyn-330)
- [Two-Dimensional Flows](https://manyspace.aporia.systems/topics/two-dimensional-flows-mth-dyn-331)
- [Bifurcation Theory](https://manyspace.aporia.systems/topics/bifurcation-theory-mth-dyn-332)
- [Chaos & Strange Attractors](https://manyspace.aporia.systems/topics/chaos-strange-attractors-mth-dyn-333)
- [Hamiltonian & Conservative Systems](https://manyspace.aporia.systems/topics/hamiltonian-conservative-systems-mth-dyn-334)
- [Dynamical Systems Applications](https://manyspace.aporia.systems/topics/dynamical-systems-applications-mth-dyn-335)
- [Electric Charge & Coulomb's Law](https://manyspace.aporia.systems/topics/electric-charge-coulomb-s-law-phy-eandm-053)
- [Electric Fields](https://manyspace.aporia.systems/topics/electric-fields-phy-eandm-054)
- [Gauss's Law](https://manyspace.aporia.systems/topics/gauss-s-law-phy-eandm-055)
- [Electric Potential](https://manyspace.aporia.systems/topics/electric-potential-phy-eandm-056)
- [Capacitance & Dielectrics](https://manyspace.aporia.systems/topics/capacitance-dielectrics-phy-eandm-057)
- [Current & Resistance](https://manyspace.aporia.systems/topics/current-resistance-phy-eandm-058)
- [DC Circuits](https://manyspace.aporia.systems/topics/dc-circuits-phy-eandm-059)
- [Circuit Analysis Methods](https://manyspace.aporia.systems/topics/circuit-analysis-methods-phy-eandm-060)
- [AC Circuits](https://manyspace.aporia.systems/topics/ac-circuits-phy-eandm-061)
- [Magnetic Fields](https://manyspace.aporia.systems/topics/magnetic-fields-phy-eandm-062)
- [Biot-Savart & Ampere's Law](https://manyspace.aporia.systems/topics/biot-savart-ampere-s-law-phy-eandm-063)
- [Faraday's Law & Induction](https://manyspace.aporia.systems/topics/faraday-s-law-induction-phy-eandm-064)
- [Inductance & Inductors](https://manyspace.aporia.systems/topics/inductance-inductors-phy-eandm-065)
- [Maxwell's Equations](https://manyspace.aporia.systems/topics/maxwell-s-equations-phy-eandm-066)
- [Electromagnetic Waves](https://manyspace.aporia.systems/topics/electromagnetic-waves-phy-eandm-067)
- [Wave Propagation & Impedance](https://manyspace.aporia.systems/topics/wave-propagation-impedance-phy-eandm-068)
- [Center of Mass & Rigid Body Motion](https://manyspace.aporia.systems/topics/center-of-mass-rigid-body-motion-phy-mech2-084)
- [Angular Momentum & Torque](https://manyspace.aporia.systems/topics/angular-momentum-torque-phy-mech2-085)
- [Non-Inertial Frames](https://manyspace.aporia.systems/topics/non-inertial-frames-phy-mech2-086)
- [Gravitation & Kepler Problems](https://manyspace.aporia.systems/topics/gravitation-kepler-problems-phy-mech2-087)
- [Constraints & Generalized Coordinates](https://manyspace.aporia.systems/topics/constraints-generalized-coordinates-phy-mech2-088)
- [Lagrangian Mechanics](https://manyspace.aporia.systems/topics/lagrangian-mechanics-phy-mech2-089)
- [Symmetries & Noether's Theorem](https://manyspace.aporia.systems/topics/symmetries-noether-s-theorem-phy-mech2-090)
- [Hamiltonian Mechanics](https://manyspace.aporia.systems/topics/hamiltonian-mechanics-phy-mech2-091)
- [Central Force Problems](https://manyspace.aporia.systems/topics/central-force-problems-phy-mech2-092)
- [Small Oscillations](https://manyspace.aporia.systems/topics/small-oscillations-phy-mech2-093)
- [Nonlinear Dynamics Intro](https://manyspace.aporia.systems/topics/nonlinear-dynamics-intro-phy-mech2-094)
- [Tensor Calculus](https://manyspace.aporia.systems/topics/tensor-calculus-phy-gr-165)
- [Differential Geometry Foundations](https://manyspace.aporia.systems/topics/differential-geometry-foundations-phy-gr-166)
- [Equivalence Principle](https://manyspace.aporia.systems/topics/equivalence-principle-phy-gr-167)
- [Curved Spacetime](https://manyspace.aporia.systems/topics/curved-spacetime-phy-gr-168)
- [Einstein Field Equations](https://manyspace.aporia.systems/topics/einstein-field-equations-phy-gr-169)
- [Schwarzschild Solution](https://manyspace.aporia.systems/topics/schwarzschild-solution-phy-gr-170)
- [Black Holes](https://manyspace.aporia.systems/topics/black-holes-phy-gr-171)
- [Gravitational Waves](https://manyspace.aporia.systems/topics/gravitational-waves-phy-gr-172)
- [Tests of GR & Cosmology Intro](https://manyspace.aporia.systems/topics/tests-of-gr-cosmology-intro-phy-gr-173)

## Resources

- Thomas' Calculus — Hass, Heil & Weir
- Calculus: Early Transcendentals — James Stewart
- Calculus: Early Transcendentals — James Stewart