Unit 1: Vectors and Euclidean Spaces

This unit covers the theory, techniques, and applications of vectors, and also covers the geometry of two- and three-dimensional space. You can visualize vectors as arrows, which reminds us that vectors have both magnitude (length) and direction. Vectors show up in many places, both in real-world applications associated with various disciplines (e.g., chemistry, physics) and in purely mathematical settings (e.g., a proof). In this unit we'll learn about vectors, their arithmetic, and how they can help us describe sets and shapes in space (like lines and planes).

Lesson 0 is a short lesson (two pages) that presents some introductory concepts and terminology, provides motivation for what we will study in this course, and discusses the overarching themes and driving question of the course.

Lesson 1 picks up that thread by studying three-dimensional space, where we'll spend most of our time in this course.

In Lesson 2 we'll introduce vectors and discuss one way to multiply them (the dot product) and the significance and applications of that.

Lesson 3 continues this approach and discusses another way to multiply vectors (the cross product) along with some real-world applications of that and an application to describing lines and planes in three-dimensional space.

Finally, Lesson 4 generalizes the description of lines via vectors to enable us to describe curves via vectors. The "vector-valued functions" that result allow us, for the first time in the course, to do calculus on vectors, which we discuss in the lesson as well.

After completing this unit we will have learned how to work with the mathematics of three-dimensional space, visualize sets and shapes in that space, and describe them with vectors. We will draw on this knowledge time and again in the subsequent units, alternating between the point view of space (space as a collection of points) and the vector view of space (space as a collection of arrows containing information about magnitude and direction at a point). For this reason, it is fair to say that this unit provides the foundational knowledge we will need to build up the rest of the mathematical edifice of this course.

This unit will also introduce, reinforce, and emphasize mathematical modeling. This is the process of translating a real-world problem into mathematics, solving it, and interpreting your solution in the original real-world context that initiated the process. You no doubt have experience doing this in calculus courses (e.g., in optimization problems). This unit will hone that skill and help you start to think more like an applied mathematician. This will, in turn, be useful for tackling your science courses and social science courses, many of which involve mathematical analyses and/or modeling.