The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Three-Dimensional Cross Product Derivation

Date: 07/26/99 at 08:45:22
From: Rohin Wood
Subject: Derivation of 3-dimensional cross product.


I am a 17 year old, year 12, maths 2 student who completely dislikes 
the idea of accepting a formula without knowing and understanding its 
derivation and working principles. So naturally when my textbook gave 
the equation for 3D cross product in terms of determinants without 
explanation and my teacher was unable to assist, I was rather annoyed.

I would be grateful for any help you could give me,
Rohin Wood

Date: 07/26/99 at 14:32:28
From: Doctor Anthony
Subject: Re: Derivation of 3-dimensional cross product.

The easiest proof uses components   a = a1.i + a2.j + a3.k
                                    b = b1.i + b2.j + b3.k
                                    c = c1.1 + c2.j + c3.k

To show  a x (b x c) = (a.c)b - (a.b)c  we have

           b x c = |i    j    k|
                   |b1  b2   b3|
                   |c1  c2   c3|

                 = i(b2.c3-b3.c2) - j(b1.c3-b3.c1) + k(b1.c2-b2.c1)

     a x (b x c) = |  i                j                 k |
                   |  a1              a2                 a3|
                   |b2.c3-b3.c2   b3.c1-b1.c3   b1.c2-b2.c1|

                 = i(a2.b1.c2-a2.b2.c1-a3.b3.c1+a3.b1.c3)   

and similar results for the j and k components.

Now compare (1) with the i component of:

     (a.c)b1 - (a.b)c1  = (a1.c1+a2.c2+a3.c3)b1 - 

          = a1.b1.c1+a2.b1.c2+a3.b1.c3 - (a1.b1.c1+a2.b2.c1+a3.b3.c1)

          = a2.b1.c2-a2.b2.c1-a3.b3.c1+a3.b1.c3   

and we can see that expressions (1) and (2) are identical. We could do 
the same for the j and k components to show that

     a x (b x c) = (a.c)b - (a.b)c

- Doctor Anthony, The Math Forum   
Associated Topics:
High School Linear Algebra

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.