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### Three-Dimensional Cross Product Derivation

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

G'day,

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.

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)
........(1)

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+a2.b2+a3.b3)c1

= 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
..................(2)

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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Algebra

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