       Re: Euler rotation angles

• To: mathgroup at smc.vnet.net
• Subject: [mg46748] Re: [mg46739] Euler rotation angles
• From: Oleksandr Pavlyk <pavlyk at phys.psu.edu>
• Date: Fri, 5 Mar 2004 01:46:58 -0500 (EST)
• Organization: Penn State University; Department of Physics
• References: <200403040547.AAA14288@smc.vnet.net>
• Reply-to: pavlyk at phys.psu.edu
• Sender: owner-wri-mathgroup at wolfram.com

```Hi Sam,

You should have been more specific about what dimensions
your vectors are. I would assume they are 3-dimensional.

Two vectors, call them v1, v2, span a plane.
Two vectors can be rotated into one another only
if they have the same length. So I will assume they do.

Within that plane, the angle between them can be
found

phi = ArcCos[ (v1.v2)/Sqrt[ (v1.v1) (v2.v2) ] ]

The perpendicular vector to that plane can be found
using cross product

perp = Cross[ v1, v2 ]

Is this vector vanishes, then your vectors are parallel.

The the sought after rotation is around vector perp through
the angle phi found above. Let us define

{nx, ny, nz} = perp/ Sqrt[ perp.perp ] ;

The matrix rot, that implements this rotation in 3 dimensions looks
as this

rot = MatrixExp[ phi { { 0, nz, -ny}, { -nz, 0, nx}, {ny, -nx, 0} } ]

rot.v2 == v1

gives True.

To understand the group of rotations in 3 dimensions better you could
consult some of books on the subject, like

"Representations of the rotation and Lorentz groups : an introduction"
by Moshe Carmeli available at Amazon (used).

http://mathworld.wolfram.com/Rotation.html

Best,
Sasha

S. Campbell wrote:

> Hi,
>
> Can anyone help me with the following? I have two vectors, one rotated
> with respect to the other, and I wish to find the Euler rotation
> angles that connect the vectors. Can mathematica tell me what the
> required angles are?
>
> Cheers,
>
> Sam

```

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