Course: Calculus III
Topic: Vectors and Geometry of Space
Subtopic: Cross Product


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.


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


Terms you should be able to define: right hand rule, cross product, vector product, norm, normal vector, determinant, adjunct, triple scalar product, parallelepiped, torque

Formulae to have in your notes:

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.

As mentioned in a previous lesson notes, CalcPlot3D is a fairly comprehensive free interactive colour 3D graphing software (Java-based) worth checking out if you need one. Detailed directions at CalcPlot3D Help Manual. (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. (In CalcPlot3D: Menu >> File >> View Vector Explorations >> in left sidebar choose "Cross Product Exploration" from the drop down box. Play around by pulling the red and blue vectors on the 2D graph to see how the cross product vector changes.)
  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

James Sousa's MathIsPower4U - Calc II:
Vector Cross Products
Find the Cross Product of Two Vectors
Find Two Unit Vectors Orthogonal to Two Given Vectors
Ex 1: Properties of Cross Products - Cross Product of a Sum and Difference
Ex 2: Properties of Cross Products - Cross Product of a Sum and Difference
Find the Area of a Triangle Using Vectors
Find the Distance Between Two Points In Space
An Application of Cross Products: Torque
The Triple Scalar Product: Volume of a Parallelepiped