Welcome!greenspun.com : LUSENET : quaternions : One Thread
Welcome to the quaternions forum, the web's [only?] forum for the discussion of quaternions and related matters, such as: physics, math, history, and why this old forgotten tool may be interesting.
-- Douglas Sweetser (email@example.com), March 25, 2002
There are the real numbers, which can be added together, subtracted, multiplied and divided. The is also the notion of how far away one real number is from another. Real numbers are hugely important in math, science, even the World economy. Complex numbers are pairs of numbers which also have these properties. They play as big a role in both math and science, and only play an economic role on Wall Street with sophisticated differential equations. There was a lot of resistance to their being accepted in the mathematics community, but the usefulness of complex numbers helped them win the day.
Can three numbers be add, subtracted, multiplied, and divided? It is the division step that makes things tricky. The answer turns out to be no. Four numbers are required to get division to work (this was proved over a hundred years ago). Gauss was the first person to figure this out, but it was Hamilton who called them quaternions.
Today, there is only one wide use of quaternions: for doing 3D rotations. All spacecraft use quaternions for this calculation because it is easier to manage errors. Otherwise, quaternions play almost no role, and are not even taught to people with a technical education.
I find it exceptionally odd that given how powerful the real and complex numbers are, quaternions play such an insignificant role. My research effort is to find out whether this island of misfit numbers should have a more central part in physics. That's the big picture.
Still, what are they? I think a quaternion represents an event, this happened at this time t, and this place x, y, z. Physics describes patterns of events, the ball flew from x, y, z at time t to a different t, x, y, z, and if all the math is done right, all the points between can be determined.
So there are two answers. Mathematically:
Real numbers (1D) -> Complex numbers (2D) -> Quaternions (4D)
Physically, any pattern of events in space and time may be described by quaternion expressions.
-- Doug Sweetser (firstname.lastname@example.org), October 17, 2002.
Using quaternions to represent (t, x, y, z) is certainly the way I view it. This is a bit of a jump however, because there is no agreement about the definition of time. The assertion that a quaternion represents (t, x, y, z) puts technical constraints on both space and time. I think those constraints are consistent with the behaviour of spacetime, but there are few on the planet that believe that.
As an example, you can find a quaternion R such that R (t, X) = (-t,X) where the capital X represents x, y, and z. There is a different quaternion, call it R' that does the reverse, R' (-t, X) = (t, X). In most situation, R is almost, but not quite R'. The widgets of time reversal are different for going backwared in time versus forward. This is not so in standard physics which uses the same member of the Lorentz group (fancy words, but a simple idea). This might help to explain why there is the second law of thermodynamics. Deep stuff, maybe.
-- Doug Sweetser (email@example.com), October 19, 2002.
(0,P)(0,Q) = (-P.Q, PxQ). Don't forget to dot your products!
-- Doug Sweetser (firstname.lastname@example.org), November 06, 2002.
Whats a quaternion?
-- JSENLIB (||@greenspun.com), October 17, 2002.
like complex numbers can be used to represent (x,y) so quaternions can be used to represent (time,x,y,z) right?
-- JSENLIB (||@greenspun.com), October 18, 2002.
Complex numbers have the form (a + bi) where i^2 = -1 ("^" = "to the power of"). They are often writen (a,b) to save space Quaternions have the general form (a + bi + cj + dk) i^2 = j^2 = k^2 = -1; ijk=-1; If you multiply both sides of the last equation on the left by i, you get -jk=-i => i=jk. Times i=jk on the left by j and you get ji=-k Times i=jk on the right by k and you get ik=-j Going back to ijk=-1, muliplying on the right by k gets -ij=-k => ij=k Times ij=k on the right by j and you get -i=kj =>kj=-i Times ij=k on the left by i and you get -j=ik =>ik=-j
From this you can see that jk=-kj; ij=-ji; ki=-ik ie quartnions do not commute.
They can be abbrivated as (a,b,c,d) or even as (a,P) where P is a vector such that P=bi + cj + dk
You can check that (a + bi + cj + dk)(w + xi + yj + zk) = (aw - bx - cy - dz) +i(ax + bw + cz - dy) +j(ay - bz + cw + dz) + k(az + by - cx + aw).
If we write the two quaternions multiplies above as (a,P) and (w,Q), and set the real parts (a and w) to zero, then the product is EXACTLY the cross product between the two vectors. This is crucial to any application of quaternions.
This should give all the maths skills you need to manipulate quaternions, assuming you can work with real numbers and vectors. (Anything else is a bonus). I hope this is fouse to you.
Tom Willis (webmaster: www.durham.ac.uk/thomas.willis PLEASE visit :-) )
-- Tom Willis (Thomas.email@example.com), November 06, 2002.
Can one interpret the electrotechnical notion "apparent power" S = UI (unit VA)as a quaternion the real part of which is "effective power" P = UI cos phi (unit W =J/s) and vector part or imaginar part is Q = UI sinphi (unit VAr) is vector product of U and I, which is orthogonal to vectors U and I? If we write quaternion S(q)= U()q I(q)building the ordinary quaternion product the effective power is allways negative. Is it possible to define a new product "quaternion circle product" I o U = UI* ( I* is the conjugate of the quaternion I, = -I because quaternion I is a pure or a vectorial quaternion), to get the right sign (+)for effective power? Is it possible to interprete the apparent power quaternion in the Minkowski┤s 4-dimensional space so that effective power P is timelike component and U and I are specelike components as Q, which is vector product of U and I and orthogonal to U and I, which vectors span the plane xy? Thankful Pekka T.
-- Pekka Tapio Laakso (firstname.lastname@example.org), January 21, 2003.
When rebuilding accepted equations using quaternions, it is essential that all the signs are correct. Most of my work has been about "getting the signs right." I am not familiar with apparent power S. If this is normally assigned positively for the dot product of U and I, take the conjugate of one or the other, S = U I* = |U| |I| cos (phi).
Personally, I avoid the word "imaginary." Historically it was used to make fun of people trying to understand complex numbers. Physically it has no meaning. I like to call the first part of a quaternion a scalar, and the next three numbers a 3-vector. One must be careful doing this since people who use tensors refer to a rank 0 tensor as a scalar and a rank 1 tensor as a vector.
I think of the scalar part of a quaternion as being "watch-related". In other words, I need my watch to figure out the amount of time that has pasted, to calculate the power, to quantify the energy. The 3-vector part is "finger-pointing-related." I use my fingers to point in the description: the event is over there, the charges are moving from here to there, and the momentum goes that way. It is all about watches and finger pointing.
-- Doug Sweetser (email@example.com), January 22, 2003.
hello, ř see your page new.an I have a question about scaler and vectorel multiplication. I am physics student. Is there anybody who know why we use in A.E= ┬.╩.Cos Q , cos Q and in AxE=┬.╩.Sin Q , Sin Q ?thank for your help ...
-- ebrar gozubuyuk (firstname.lastname@example.org), June 02, 2003.