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Re: Dot Product in Cylindrical Coordinates
*To*: mathgroup at smc.vnet.net
*Subject*: [mg69276] Re: Dot Product in Cylindrical Coordinates
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Tue, 5 Sep 2006 05:30:40 -0400 (EDT)
*Organization*: The University of Western Australia
*References*: <edadqd$pgc$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <edadqd$pgc$1 at smc.vnet.net>,
Sergio Miguel Terrazas Porras <sterraza at uacj.mx> wrote:
> When I calculate the dot product of vectors {1,Pi/4,0} and {2,0,1} in
> Cylindrical Coordinates Mathematica 5.1 returns the result Sqrt[2], when the
> result should be 2.
Notwithstanding several of the other responses, the result _is_ Sqrt[2].
When you write {1,Pi/4,0}, surely you mean
{rho, phi, z} == {1, Pi/4, 0}
and _not_ that
{x, y, z} == {1, Pi/4, 0} ?
After loading
Needs["Calculus`VectorAnalysis`"]
and selecting Cylindrical coordinates,
SetCoordinates[Cylindrical];
then in cartesian coordinates, this point is
p1 = CoordinatesToCartesian[{1, Pi/4, 0}]
{1/Sqrt[2], 1/Sqrt[2], 0}
Similarly,
p2 = CoordinatesToCartesian[{2, 0, 1}]
{2, 0, 1}
Hence the dot product of the coordinate vectors (relative to the origin
{0,0,0}), computed in cartesian coordinates, is
p1 . p2
Sqrt[2]
This is the same result that you got, presumably using,
DotProduct[ {1, Pi/4, 0}, {2, 0, 1} ]
Sqrt[2]
Of course, if you really mean
{x, y, z} == {1, Pi/4, 0}
then there is no need to load Calculus`VectorAnalysis`: the dot product
is just
{1, Pi/4, 0} . {2, 0, 1}
2
Note that Dot is _not_ modified when this package is loaded so
Jean-Marc's response,
Needs["Calculus`VectorAnalysis`"]
SetCoordinates[Cylindrical];
{1, Pi/4, 0} . {2, 0, 1}
is bogus -- the first two lines have no effect on the third.
Modifying Andrzej's code, we have
Simplify[(JacobianMatrix[] . p1) . (JacobianMatrix[] . p2)]
Sqrt[2]
Cheers,
Paul
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
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