Dave4Math / Courses / Vectors and the Geometry of Space

Vectors and the Geometry of Space

Master vectors and the geometry of space. You'll learn about vectors in 3-space, the dot and cross products, lines, planes, and surfaces in space in this course.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Have you ever been in a math class and said to yourself: “Wow, I like this more than I thought I would”? Introducing Dave’s online math course: Vectors and the Geometry of Space. Don’t just learn, be inspired! So you too can have confidence.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in vectors and the geometry of space.
  • Students who are enrolled in calculus 2 and want to improve their grade.
  • Anyone interested in getting prepared for calculus 3.
  • Anyone wanting to learn about vectors in 3D, the dot product and cross product in 3D, various 3D coordinates systems, and lines, planes, and surfaces in 3D.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Vectors and the Geometry of Space

  • Vectors
  • Scalar Multiples
  • Vector Addition: The Parallelogram Law
  • Vectors in the Coordinate Plane
  • Length of a Vector
  • Vector Addition in the Coordinate Plane
  • Properties of Vectors
  • Unit Vectors
  • Standard Basis Vectors
  • Angular Form of the Unit Vector
  • Coordinate Systems in Space
  • The Distance Formula
  • The Midpoint Formula
  • Vectors in 3-Space
  • Standard Basis Vectors in Space
  • Finding the Dot Product
  • The Angle Between Two Vectors
  • Orthogonal Vectors
  • Direction Cosines
  • Vector Projections and Components
  • Work
  • The Cross Product of Two Vectors in Space
  • Geometric Properties of the Cross Product
  • Finding the Area of a Triangle
  • Properties of the Cross Product
  • The Scalar Triple Product
  • Equations of Lines in Space
  • Equations of Planes in Space
  • Parallel and Orthogonal Planes
  • The Angle Between Two Planes
  • The Distance Between a Point and a Plane
  • Traces
  • Cylinders
  • Quadratic Surfaces
  • The Cylindrical Coordinate System
  • The Spherical Coordinate System

Course Description

In this course, we begin with an introduction to vectors. We discuss vectors in the plane, and we go into detail on vector addition, scalar multiplication, and the length of a vector. Many properties of vectors and unit vectors are also covered.

After this introduction to vectors, we begin working in three dimensions. Many of the concepts just introduced are implemented easily in 3-space. We describe the dot product and cross product, including vector projections, work, and finding areas. Many properties of both these operations are covered in detail with several examples.

After these studies on vectors, we discuss lines and planes in space. We thoroughly explain equations of lines in space, equations of planes in space, and several other topics such as the angle between two planes and the distance between a point and a plane.

After studying vectors, lines, and planes, we now investigate surfaces in space. We cover cylinders and various types of quadratic surfaces, including ellipsoids, hyperboloids, cones, and paraboloids.

Having explored rectangular coordinates so far in this course, we now turn our attention to other coordinate systems. The cylindrical coordinate system and the spherical coordinate system are detailed. We explain how these coordinate systems work and how to convert points and equations between them, including rectangular coordinates.

Recommended Prerequisites for Vectors and the Geometry of Space

I recommend the prerequisite course Conic Sections, Plane Curves, and Polar Coordinates. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 2 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.