Algebra I / Elem. Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems
Algebra II / E&I Algebra
Exponents & Polynomials Intermediate Algebra starts here!

Factoring Rational Expressions Rational Equations and Applications
Algebra III / Inter. Algebra
Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III
Sequences and Infinite Series Power Series Vectors and Geometry of Space Vector-Valued Functions
Calculus IV
15.Functions of Several Variables
16.Multiple Integration
17.Vector Analysis
S = contains supplemental resources
Course: Calculus III
Topic: Vectors and Geometry of Space
Subtopic: Cross Product

Overview

A cross product is one method of multiplying two 3D vectors. The operation is written with a cross (×) between the vectors such as <1,2,3>×<4,5,6>. While the answer to a dot product is a constant, the answer to a cross product is a vector, thus it is also known as a vector product. The answer vector is perpendicular to the plane spanned by the two original vectors. The orientation of the answer vector (ie. pointing up or down from the plane) is determined by the right hand rule. The cross product is used in vector analysis in a variety of STEM applications.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 13.4.1 Find the magnitude of the cross product of two vectors in three dimensions using the norm of the cross product formula
• 13.4.2 Determine if two vectors are parallel via cross product
• 13.4.3 Find the area of a parallelogram via cross product formula
• 13.4.4 Understand a geometric interpretation of cross product
• 13.4.5 Perform vector operations (eg. finding the magnitude) on the vector resulting from a cross product
• 13.4.6 Understand and be able to apply the Properties of the Cross Product rules
• 13.4.7 Understand all nine cross products of the basic unit vectors algebraically and geometrically
• 13.4.8 Find the cross product of two vectors in three dimensions using the determinant form including evaluating the determinant algebraically via the adjunct expansion
• 13.4.9 Find the area of a triangle via the magnitude of the determinant form of cross product
• 13.4.10 Find the volume of a parallelepiped via the absolute value of the triple scalar product
• 13.4.11 Use cross product to perform vector analysis particularly with real-life applications in STEM fields including calculating torque

Terminology

Define: cross product, vector product, norm, normal vector, determinant, adjunct, triple scalar product, parallelepiped, torque

Formulae to have in your notes:

• The determinant method of finding the cross product
<a,b,c> x <d,e,f> = det[ [i,j,k] [a,b,c] [d,e,f] ].

• The norm of the cross product ||u × v|| = ||u||·||v||·sinθ
where θ is the acute angle between vectors u and v.
(Note: if u⊥v then sin(θ)=1 and ||u × v|| = ||u||·||v||.)

• The absolute value of the triple scalar product
|u • (v × w)| = ||u||·||v × w||·cosθ
where θ is the acute angle between vectors u and v×w.

• The distributive law for cross products:
w × (u + v) = (w × u) + (w × v) and
(u + v) × w = (u × w) + (v × w)

Supplemental Resources (recommended)

Can your calculator perform Determinants, Dot Products, and Cross Products? See Prof. Keely's Calculator Guide: Determinants and Calculator Guide: Dot & Cross Products for steps on the Ti-84/86, TI-89, and HP-48.

CalcPlot3D is a nice little *free* interactive colour 3D graphing software (Java-based) worth checking out if you need one. More at CalcPlot3D Info. (Note that MAC computers already have a built-in 2D/3D grapher.)

Explore the geometric interpretation of cross product via one of the following interactive sites:

1. Explore CalcPlot3D's Cross Product Exploration (Java-based).
2. Go to Math Insight's Cross Product article. It's a pretty good read all around but in particular scroll down and play with the GeoGebra applet.
3. Play with the Wolfram Demonstrations Project: Cross Product of Vectors.

Supplemental Resources (optional)

Better Explained's article Cross Product

Paul's OL Notes - Calc II: Cross Product

Patrick JMT Just Math Tutorials:
Cross Product and Torque: An Application of Cross Product