Module 2: Introduction

In this module you will complete the following activities:

Compute the dot product of two vectors in $R^{n}$ $\mathbb{R}^{n}$ .
Compute the cross product of two vectors in $R^{3}$ $\mathbb{R} ^{3}$ .
Recall formulas which relate the dot or cross product to the angle between two vectors.
Apply these formulas to determine if two vectors are perpendicular.
Apply these formulas to compute areas of parallelograms and volumes of parallelpipeds.
Create and resolve algebraic statements involving the dot and/or cross product and their geometric properties.
Practice the covered concepts through some exercises, called "Now You Try".
Complete a quiz on this topic at the end of the module.

Introduction

Recall that the essential operations of vectors in $R^{n}$ $\mathbb{R} ^{n}$ are vector addition and scalar multiplication. In this module we will discuss two product operations: the dot and cross product. These operations are useful because they tell us information about the geometric relationship between two vectors. In particular: how "close'' two vectors are to being parallel or perpendicular in direction. Interestingly these geometric questions come into play in a variety of situations such as how much work the wind does blowing a leaf across the lawn, or how much rain might pass through an open window, or (perhaps most importantly) finding the closest approximation under some constraint.