flipcode - Sarrus Scheme for Cross Products and Determinants

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Sarrus Scheme for Cross Products and Determinants
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Sarrus scheme aka: Sarrus Rule, Rule of Sarrus, The Sarrus Expansion, aka The Enigmatic Sarrus's 12 steps to better Cross products

named for P.F. Sarrus but previously created by Takakazu Seki

Here is a little tip for anyone who ever has trouble remembering how to calculate a cross product. I just picked it up while reading through the appendix of Real-Time Rendering by Akenine-Moller and Haines.

let:


 i  be the X axis,
 j  be the Y axis,
 k  be the Z axis.

given vectors A and vectors B

A = <Ax,Ay,Az>
B = <Bx,By,Bz>

list a matrix that looks like this:

i  j  k 
Ax Ay Az
Bx By Bz

now write it twice side by side:

+  +  +      -  -   -
 \  \  \    /  /  /
  i  j  k  i  j  k   
  Ax Ay Az Ax Ay Az
  Bx By Bz Bx By Bz

then follow the + and - along the diagonals to add up the terms:

 i*Ay*Bz  +  j*Az*Bx  + k*Ax*By  - i*AzBy - j*AxBz - K*AyBx

gather the terms:

i(AyBz - AzBy)  +  j(AzBx - AxBz) + k(AxBy - AyBx)

the stuff inside the parenthesis are the components of your new vector

A cross B =  < (AyBz-AzBy), (AzBx-AxBz), (AxBy-AyBx) >

This same technique can be used to compute the determinant too. I'm sure this is an old trick to some of you, but it's new to me.

may you never forget how to compute a cross product again.
- Clint Brewer

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