       Re: Re: Wrong behavior of CrossProduct

• To: mathgroup at smc.vnet.net
• Subject: [mg8046] Re: [mg7976] Re: [mg7958] Wrong behavior of CrossProduct
• From: seanross at worldnet.att.net
• Date: Sat, 2 Aug 1997 22:32:54 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```Sergio Rojas wrote:
>
>  Hello:
>
>  How Mathematica implement the Cross Product of two vectors?
>
>   As far as I know the basic definition of the Cross Product between
> two vectors in ANY ORTHOGONAL coordinated system is as follows:
>
> (a1,a2,a3)X(b1,b2,b3) = (a2*b3 - a3*b2,
>                          a3*b1 - a1*b3,
>                          a1*b2 - a2*b1)
>
>   In physics this is usually illustrated by taking any three
>   UNIT vectors (u,u,u) with the orthogonal property:
>
>    u[i].u[j] = Delta[i,j]  where Delta[i_, j_] := If[i==j, 1, 0]
>
>    uxu =  u ; uxu =  u ; uxu =  u
>    uxu = -u ; uxu = -u ; uxu = -u
>    uxu =   0   ; uxu =   0   ; uxu =   0
>
>   Then, the above result follows by expanding:
>
> (a*u + a*u + a*u)x(b*u + b*u + b*u)
>
>  and using the orthogonal property of the unit vectors.
>
>   In my example,
>
> a = ( 0, 0,1) ; b = (a1,a2,0)
>
> axb = ( a2, -a1, 0 ) .

No, you are forgetting the metric tensor, which for cartesian
coordinates is the identity matrix and has non-zero diagonal elements
that are a function of r and theta in spherical coordinates.  Even some
I think the failure to mention little details like this is that most
physics books are written by theoreticians who rarely calculate
numbers-they just generate formulas and let somebody else actually
calculate the numbers.  Anyway, the components of the metric
tensor(usually written g) are the square of the scale factors(usually
written h) for the particular component.  As I recall, for cylindrical
coordinates g11=1, g22=1/r^2, g33=1 and for spherical g11=1, g22=1/r^2,
g33=1/r^2 sin(theta)^2, but don't quote me on that.  I got sick of
theoreticians and became an engineer some time ago.

```

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