Module 1: Introduction
In this module you will complete the following activities:
- Express a vector as a (vertical) list of n real numbers.
- Identify the dimension of a vector and its components.
- Add vectors componentwise.
- Multiply a vector by a scalar.
- Compute the magnitude of a vector and the distance between two vectors.
- Create and resolve algebraic statements using the algebraic operations of vector addition, scalar multiplication, and magnitude/distance.
- Practice the covered concepts through some exercises, called "Now You Try".
- Complete a quiz on this topic at the end of the module.
Introduction
Nearly all mathematics you've done so far has been with functions of single variable and applied to one dimensional models. While this is a valuable simplification for learning the essentials of, say, calculus; it is true that most interesting real world problems are multidimensional in nature. For example, one's location on a map requires longitude x and latitude
y. Note that your location requires two measurements which a mathematician might interpret as two dimensions of information (the point
(x,y)). Suppose further that as you travel around town you also want to keep track of when you were in a particular place. Including time (
t) now makes the scenario three dimensional:
(x,y,t). Maybe you also want to know your linear velocity
(V) at each moment. Now the complete information of your location, when you were there, and how fast you were traveling is four dimensional:
(x,y,t,v).
Even this example is rather trivial. Solving modeling problems with hundreds of thousands (even millions!) of dimensions are done regularly these days. In order to handle such large problems, mathematicians rely on the concept of a vector to manage otherwise unwieldy amounts of data. The purpose of this module is to acquaint yourself with the most basic type of vector (a list of real numbers) and the essential arithmetic and algebraic operations you should be familiar with.