TimeMaster 1a:
Thanx for the response. Let’s proceed. Point by point if you will.
You said:
<You said:[BR>“Notice how in your example “velocity” is actually a quantity OF a vector, whereas the Vector itself is the definition of direction+speed, not velocity itself.”]
That is correct. I did.
And you say…
<“I see no conflict with the above statement and the two statements that follow, since displacement is a vector quantity, and velocity is the rate of displacement.”>
I do, but…OK.
And then…
<“Quantities which require both a magnitude and a direction for their complete description are called VECTOR QUANTITIES. Therefore VELOCITY is a vector quantity.”>
I agree, but then now I’m confused as to what it is YOU are saying since the above statement is in accordance with the quote I provided (from Harcourt), yet seems to be at odds with your ORIGINAL STATEMENT that as follows: (and please correct me if I am wrong)
’ "“The quantity in physics which combines the speed of a body with the direction in which it moves is called velocity.” ’
Sorry, but you have baffled me here with a seeming contradiction. In YOUR quotes.
(I thought I was the one proporting that velocity is a vector quantity, not the state of a vector as YOU purport. Based on the evidence I provided of course.)
And then:
<“The speed of an object indicates how fast it is moving, that is, the DISTANCE the object will travel in a given time. It tells nothing, however, about the direction in which the object is moving.” [P>“The quantity in physics which combines the speed of a body with the direction in which it moves is called velocity.”
These two above statements are direct quotes from my Physics text book, therefore I feel compelled to accept them as accurate.]
Ok. I’ll accept that YOU do. And then ask how we can the resolve the difference between your Physics textbook and the Harcourt Dictionary, as far as the SECOND statement above is concerned.
After all, we can’t even agree to disagree if we use reputable sources that conflict with each other now can we. 
You said:
“By the way, Time Dilation has NOTHING to do with distance at all. The Time Dilation itself is a DIRECT Result of the difference in RELATIVE velocity of any two given bodies.”
Yup, I said that. But I will go a step further.
In the above case, I’m referring to Time Dilation in the original sense as it was discovered and predicted by Einstein’s first disertation on it. Namely the form of it that DOES indeed pertain ONLY to Velocity.
(There are SEVERAL forms of the manifestation of Time Dilation. I was in this case referring to only ONE of them. I can even stand corrected if you like, but if you will, please forgive the single mindedness of this particular focus as to which manifestation.)
I concede that there is another and even contradicting form of Time Dilation that relates to gravity which indeed WOULD be affected by Distance, but perhaps shortsightedely, I was merely referring to but one of them here, since the two are not directly related. They are actually opposing forces, not coincidental ones.
The thing here is, that the gravity induced form of Time Dilation is so much weaker, so much less in significance IN COMPARISON TO the velocity induced form, that it’s effect on the velocity induced form is almost entirely negligible.
Remember the scenario I used where two people walking away from each other do induce Time Dilation, but in comparison to the Space Shuttle vs. The Earth, the human scenario is negligble.
So please allow me to modify the statement just a tad by saying, that Distance has ALMOST nothing to do with it. Certainly compared to velocity anyway.
Then YOU said:
“To say that Time Dilation does not exist as a function of DISTANCE is like saying that the root of a tree has nothing to do with the growth of the leaves.”
Well, OK given the above concession. But in comparison, I still stand by the original statement in a very general sense.
And then you said:
"It is not logical to say “The Time Dilation itself is a DIRECT Result of the difference in RELATIVE velocity of any two given bodies.” ’
Of course it is.
Because it is true!
As predicted.
And measured and verified by repeatable experiment.
(and further…)
“and at the same time deny that it is the RELATIVE distance between any two bodies that is at the root of Time Dilation.”
You’re mixing Apples and Oranges here.
Dilation as a result of relative velocity has NOTHING to do with distance. And I stand behind that statement.
But to clarify, I’ll agree that Dilation due to gravity does. But it’s like licking raindrops in a rainstorm trying to catch all the water falling from the sky.
I would speculate however that an individual at the event horizon of a Black Hole observing a traveler moving away from him at the speed of light would constitute two guys that could communicate in real time with each other (given the technology to do so), since in THAT one particular scenario, the dilation would be cancelled between them.
Peace.
This is good.