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Question about Point

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What does it mean to divide a point by another point? I have this intersection problem :
Given the points P0 = (1.78; 2.13); P1 = (9.55, 18.47);Q0 = (2.11, 1.33);Q1 = (7.77, 22.19) and the following equations for two lines :

P(t) = (1 - t)P0 + tP1, where 0 <= t <= 1

Q(s) = (1 - s)Q0 + sQ1, where 0 <= s <= 1
By setting them equal to each other and letting s = t, I get the following formula:
t = (Q0 - P0) / (deltaP - deltaQ), where deltaP = P1-P0 and deltaQ = Q1-Q0.
after that computation I got :
Tx:  0.15639806
Ty:  0.17699115
plugging these values into P(t) and Q(s) I get the following result :
Q(t), x =  2.995213
Q(t), y =  5.0220356
Q(s), x =  2.995213
Q(s), y =  5.0220356
So it seems that it effectively found the intersection point, but to me it does not make sense, since we are dividing a Point by another Point. Can someone shed some light on this ?

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The reason is because I wanted to find a single T such that P(T) = Q(T).
If there is some intersection, its going to happen at some time t, meaning
that an intersection can only occur at the same time. Its kind of iffy statement,
but right now I can't word it correctly.

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It's an incorrect statement. Consider P0=(-1,0), P1=(1,0), Q0=(0,-1), Q1=(0,3). Clearly these two lines intersect at (0,0), but the parameters are different: 0.5 and 0.25. Remember, "t" in the parametric equation of a line doesn't mean "time".

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Ya I see that. I am not completely sure though, but it worked out. Thats why I'm
hoping someone can explain this. I'll think about it more in daytime, tomorrow.

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Dividing a point by another point makes no sense, unless some sort of multiplication of points is defined, as in an elliptic curve.

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