Unit 1 of 6

Vectors and the Geometry of Space

  • 7 lessons
  • Parts 1 to 14

Multivariable calculus happens in space, so it starts with a precise way to describe space. This unit sets up right-handed coordinates in three dimensions, then builds vectors with their lengths, directions and components, and the dot and cross products that measure angles, projections, areas and volumes.

Matrices and determinants describe linear maps, which return throughout the course as derivatives. The unit then writes equations of lines and planes, sketches cylinders and quadric surfaces from their traces, and introduces cylindrical and spherical coordinates.

Every lesson is a pair of videos. The theory part explains the idea and the practice part works through problems. Each lesson page has the definitions and an example for every key term.

Coordinates, Vectors, and Products

  1. Three-Dimensional Coordinates and Vectors

    How to locate points in right-handed three-dimensional space, measure distances, write spheres, and describe displacements and forces as vectors with components.

    Theory part 1Practice part 2

  2. The Dot Product

    How one number built from two vectors measures lengths, angles, perpendicularity, projections and the work done by a constant force, and why it can never exceed the product of the lengths.

    Theory part 3Practice part 4

  3. The Cross Product

    How two vectors in space produce a third vector perpendicular to both, how its right-handed direction and its length give normals, areas and torque, and how the triple product measures volume.

    Theory part 5Practice part 6

Linear Maps, Lines, and Planes

  1. Matrices, Linear Maps, and Determinants

    How a matrix acts as a linear map, how products compose maps and solve systems, and how the determinant measures area or volume scaling and orientation.

    Theory part 7Practice part 8

  2. Equations of Lines and Planes

    How to write lines in vector, parametric and symmetric form and planes through a normal vector, and how to compute their intersections, angles and distances.

    Theory part 9Practice part 10

Surfaces and Coordinate Systems

  1. Cylinders and Quadric Surfaces

    How to picture a surface in space from its traces, and how to recognize cylinders and the standard quadric surfaces from the signs and shifts in their equations.

    Theory part 11Practice part 12

  2. Cylindrical and Spherical Coordinates

    How cylindrical and spherical coordinates locate a point, how to convert points, surfaces and solids with the right quadrant and domain, and where the coordinates stop being unique.

    Theory part 13Practice part 14