Re: vector directions

["John S. Denker" <jsd@AV8N.COM>]

On 10/04/2003 02:28 PM, Michael Edmiston wrote:
> ... that an xyz coordinate system has mutually perpendicular axes,
> that we choose to use the right-handed version (such that i cross j =
> k), that angles in the x-y plane are measured counterclockwise from
> the x axis when the origin is viewed from a position on the positive
> z axis

Let's compare and contrast that with the following
geometric versions:

x and y are orthonormal.  (That is, they are unit
vectors and are mutually perpendicular.)  A small positive
rotation in the xy plane will take something that was
initially in the +x direction and rotate it toward the
+y direction.

x, y, and z are orthonormal.  (That is, they are unit
vectors and are mutually perpendicular.)  A small positive
rotation in the xy plane will take something that was
initially in the +x direction and rotate it toward the
+y direction.

x, y, z, and t are orthonormal.  (That is, they are unit
vectors and are mutually perpendicular.)  A small positive
rotation in the xy plane will take something that was
initially in the +x direction and rotate it toward the
+y direction.

Discussion:

 *) All three geometric versions are verbatim identical
starting with the first verb ("are").

 *) This way of describing rotations works the same in
two, three, or four dimensions, which is obviously a
good thing.

 *) We can describe rotations without mentioning the
axis of rotation -- instead we routinely talk about the
plane of rotation and leave it at that.

 *) We can describe rotations without mentioning the cross
product.

 *) Since we don't need the cross product we don't need any
right-hand rule.  There is no need to arrange for xyz to be
right-handed.  Except for things like the weak nuclear
interaction, the laws of physics don't care about handedness.

 *) For the same reason we don't need any notion of CW or CCW.
It suffices to speak of the xy direction of rotation (which
is the opposite of the yx direction).

 *) When we speak of the xy plane, xy is not an arbitrary
name;  it denotes the geometric product of x with y.  Since
x and y are perpendicular this is identical to the wedge
product x /\ y.

For present purposes geometric algebra is synonymous with
Clifford algebra.

http://www.av8n.com/physics/rotations.htm
http://www.av8n.com/physics/pierre-puzzle.htm

=========================

I really recommend the geometric way of describing rotations.
I find it works great for people at all levels, starting
with folks who don't understand physics, not even at the
high-school physics level.  I find there is nothing gained
by talking about the axis of rotation, and nothing lost by
talking about the plane of rotation.

Down with cross products!