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**

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:

- The determinant method of finding the cross product

`<<a,b,c>> xx <<d,e,f>> = |[hati,hatj,hatk],[a,b,c],[d,e,f]|`

- `vecu` and `vecv` are parallel IFF `vecu xx vecv = 0`.

- The norm of the cross product `||vecu xx vecv|| = ||vecu||*||vecv||*sintheta`

where θ is the acute angle between vectors u and v.

(Note: if `vecu_|_vecv` then `sintheta=1` and `||vecu xx vecv|| = ||vecu|| * ||vecv||`.)

- If vecu and vecv are two sides or a parallelogram, then its `"area" = ||vecu xx vecv||`.

- The absolute value of the triple scalar product

`|vecu`**•**`(vecv xx vecw)| = ``||vecu|| * ||vecv xx vecw|| * costheta`

where θ is the acute angle between vectors u and v×w.

- The distributive law for cross products:

`vecw xx (vecu + vecv) = (vecw xx vecu) + (vecw xx vecv)` and

`(vecu + vecv) xx vecw = (vecu xx vecw) + (vecv xx vecw)`

**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:

- 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.)
- 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.
- 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