direction cosines

[AntonioLao]

For an indexed angular variable [math]\theta_i[/math], known in quantum field theories as the phase, the direction cosines of a given vector [math]\mathbf{r}_i[/math] is given by [math]l_i=\cos \theta_i[/math] and [math]m_i=\sin \theta_i[/math]. Therefore, every non-zero indexed vector [math]\mathbf{r}_i[/math] is associated with a pair of these direction cosines. Although the polar variables [math]\theta_i[/math] and [math]\mathbf{r}_i[/math] are independent of each other, as shown by separability theorem http://en.wikipedia.org/wiki/Separation_of_variables, when the absolute magnitude of [math]\mathbf{r}_i[/math] vanishes (for zero or null vectors), the pair of direction cosines also vanishes. However, parallelism and perpendicularity theorem indicate that these can be used to demonstrate the existence of parallel displacement and orthogonal vectors including their spatio-temporal orientations. For the same orientation the sum of the corresponding product of direction cosines is unity, [math]l_1l_2+m_1m_2=1[/math]. For opposite orientation, [math]l_1l_2+m_1m_2=-1[/math]. For orthogonal vectors, [math]l_1l_2+m_1m_2=0[/math]. Furthermore, the triangular area bounded by three vectors is given by
[math]A=\frac{1}{2}\left|\begin{array}{ccc}x_1&y_1&1\\x_ 2&y_2&1\\x_3&y_3&1\end{array}\right|[/math]
Where [math]x_i=r_i\cos \theta_i[/math] and [math]y_i=\sin \theta_i[/math]. This area vanishes if [math]r_1=0[/math] or [math]r_2=0[/math] or [math]r_3=0[/math]. Moreover, a necessary condition for area to exist is [math]\theta_1+\theta_2+\theta_3=180^{\circ}[/math].

[Guille]

Antonio,

You are assuming that space-time is a sort of cartesian coordinate system. Why should time have it's independent axe whiles all the other dimensions share one? Time is just one other dimension, with nothing special. I believe there must be three dimensions of time but we are so stupid we don't percieve them (which doesn't mean we won't develop a method to detect them), such that each of the axes represents three dimensions (x for 3 spaces, y for 3 times).

[Lloyd Gillespie]

Antonio,

You are assuming that space-time is a sort of cartesian coordinate system. Why should time have it's independent axe whiles all the other dimensions share one? Time is just one other dimension, with nothing special. I believe there must be three dimensions of time but we are so stupid we don't percieve them (which doesn't mean we won't develop a method to detect them), such that each of the axes represents three dimensions (x for 3 spaces, y for 3 times).

I know of two dimensional time Guille, so I should suspect there is three dimensions of time...

regards

[AntonioLao]

assuming that space-time is a sort of cartesian coordinate system. Why should time have it's independent axe

On the contrary, if the Cartesian system exists then the area does not exist. I chose the existence of area at macroscopic level and the existence of orthogonal system of null vectors at the microscopic level. When these vectors are truly equal and orthogonal, all areas disappear, the outcome is quantum vacuum of space-time of one dimension.

[Lloyd Gillespie]

On the contrary, if the Cartesian system exists then the area does not exist. I chose the existence of area at macroscopic level and the existence of orthogonal system of null vectors at the microscopic level. When these vectors are truly equal and orthogonal, all areas disappear, the outcome is quantum vacuum of space-time of one dimension.

On the other contrary Antonio, the mind could not separate itself from motion unless it had a permanent stationary time space to function from, to witness the very movement of motion___this is the old logical philosophical point about the necessity of dual-nature time___not Einstein's falsely conjoined relative space-time. If you have a way to explain how the mind can witness motion, when it is locked in one dimensional motion, I'd like to hear it... This is the old pre-Einstein argument about time___to me, it still applies... This is where Einstein's relativity breaks down to true Newtonian philosophical reality... It's kinda like the mind/body problem, possibly it is part of the mind/body problem...

IMO, it takes a stationary mind time-space to witness all or any motion...

regards

[AntonioLao]

If you have a way to explain how the mind can witness motion

I still think that mind cannot perceive motion unless matter exists.

[Lloyd Gillespie]

I still think that mind cannot perceive motion unless matter exists.

I didn't say anything about matter not existing to perceive motion, of course it does.

regards

[AntonioLao]

didn't say anything about matter not existing to perceive motion

The mind exists to perceive everything.