Module 1 - Vectors
Introduction
The first module of this course covers vectors, and consists of four topics. Because this is the first module of a math course, and I have not taken math in a long time, I found it helpful to do extra practice during this module to get used to doing math again. I suggest anyone who is struggling or has not done math in a while do the same, as the course textbook includes gret examples but repetition is the best way to develop skill with solving math problems. I left the practice generators at the end of this page, but feel free to check them out early.
Topic 1: Geometry and Algebra of Vectors
The first topic covers a broad range of fairly simple topics. In case you struggle with what is found in our textbook, these resources may help. I also recomend using the practice generators and techniques at the end of this page to practice and get back in the flow of doing math.
How do you add points to vectors?
Khan Academy - Vectors
Linear Combinations - LibreTexts
Linear Combinations and Span - Khan Academy
Modular Arithmetic - Brilliant
Topic 2: Length and Angle
The second topic builds on the first and gives us a solid foundation for later learning. Be very careful to remember this stufff, it comes in handy later. Also, take this an an opportunity to try out Wolfram Alpha and Symbolab to see what they can do. They can give you proofs for the inequalities and theorems, calculate vector projections, angles, lengths, and more.
Vector Dot Product and Length - Khan Academy
Cauchy-Schwarz inequaltiy proof - Khan Academy
Triangle Inequality in Vectors - Cuemath
Vector Normalization - Khan Academy
Angle Between Two Vectors - Cuemath
Angle Between Two Vectors - Symbolab Calculator
Distance Between Vectors Practice Problems - Varsity Tutors
Distance Between Vectors - Andrew Misseldine
Orthogonality, Pythagorean theorem, Cauchy-Schwarz, and Triangle Inequality - Interesting read! - LibreTexts
Vector Projections - LibreText
Symbolab - Vector Projection Calculator
Topic 3: Lines and Planes
For this unit, I would recommend doing lots of practice both in this course and in questions that you generate or find for yourself. Memorize the formulas in our course text and understand how you can find alternate ways to solve the problems.
Write the equation of a line in general form, vector form, or parametric form - Engineer4Free
Lines and Planes in R3 - Harvard
Equations of Planes - Paul's Online Notes
Normal Vector From Plane Equation - Khan Academy
Lesson Explainer: Equation of a Plane: Vector, Scalar, and General Forms - Nagwa
Topic 4: Cross Product
While searching for resources relating to the cross product, I found that the cross product concept can extend far beyond the contents of this course. Fortunately, this course only seems to cover the cross product in R3, which is relatively simple. I found the course text to be mostly sufficient for this one, but here's some other resources I found:
Cross Product - University of Manitoba
Cross product calculator - Wolfram Alpha
Practice Question Generators
For this Module, I found it useful to do extra practice to help memorize key concepts and get my algebra back up to speed. Wolfram and Symbolab both do vector addition/subtraction, scalar multiplication, and dot product, but wolfram has cross products and symbolab has angles and projections. You can also input problems into either site to generate solutions for other practice questions:
Symbolab Practice - Vectors
Wolfram Alpha Problem Generator - Vectors
Brilliant - Various topics
To create practice problems in topics not included in the practice problem generators, you can use the rand button on your calculator, your computer, or your imagination to get random numbers, use these to fill out points/lines/etc, and make your own practice problems. Calculate the distance between points/planes/lines etc made using these numbers, and then ask Wolfram Alpha or Symbolab to solve the problem for you. Check your answers and if you need to then you can get a subscription to one of these services to see how to solve them.