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Now er ...
Just a few words about the objects that these three things describe.
When I began talking about geometry I [...] a line.
A line as a geometric object, as a curve.
And then algebraically we came up with ...
... system of equations.
And then physics way we came up with description terms of vectors.
Er ... now ...
Do we have one to one correspondence?
Is it that given a system we have exactly one line described by that system?
Do you have certainty about that?
If I write down a system for you ...
... with some coefficients ...
... those determine a line. Well of course they determine an equation ...
... and you convert it to a line and do you get just one object?
Now given a line ...
... do you expect to have one system describing that line?
No.
So different systems can describe exactly the same geometric object.
Because speaking of the physics understanding ...
... the motion can begin at different points.
And you will still have the same line as long as velocity vector is the same.
Or even velocity vector can be longer or shorter.
You can move along the same line faster or slower, one way or the other.
So that means algebraic and physics interpretations ...
... can change giving exactly the same geometric object.
So ...
Geometry provides ...
... probably more general concept of a geometric object ...
... then algebra and physics ...
... talking about very very specific motion along that object.
And speaking of physics ...
... the [...] or that line is going to be the trajectory.
Right? That motion is about motion of the particle that moves.
But the concept we keep in mind geometrically speaking is the trajectory of that particle.
All right. So that's what it means to be linear from different points of view.
And that means we now understand ...
... among all the curves the simple type, the simple class of curves is linear curves.
And well, by the way the time t may take negative values.
And I hope everybody is comfortable with t being negative.
Although the point begins here and moves to the right.
All the points to the left should be described by this equation.
If you see t is negative number ...
... it means something happened in the past.
The particle is presently here, in the past it was there, in the future it will be there.
So t means ...
t being a negative number still means something.
So it's not something strange as negative area, that I brought to this class.
It is something you should be comfortable with.