Parallelism of Time-varying Vectors

  • #1
236
21

Homework Statement


This is a solved problem, but I haven't understood a few things.
I've marked out sections of the solution in white for convenience. The markings are positioned where that particular section ends.
Untitled__1532764784_116.75.182.31.jpg

In part (1), how did they just assume
f1(0) = 2, f2(0) = 3, g1(0) = 3, g2(0) = 2
f1(1) = 6, f2(1) = 2, g1(1) = 2, g2(1) = 6

And, in part (4), what is this 'intermediate value theorem' that they've used?
We've just done the basics on vectors, so I have no idea where this 'intermediate value theorem' came from....
Then, in part (2) they say we have to prove that
f1(t).g2(t) - f2(t).g1(t) = 0
And in part (4), they just implied the same thing from out of nowhere & voila! The problem's over!
Please explain that too...

 

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Answers and Replies

  • #2
FactChecker
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Your objections are valid. These values seem to come out of nowhere.

The intermediate value theorem simply says that if a continuous function has two values, then it must also have every value in between the two.
 
  • #3
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The intermediate value theorem simply says that if a continuous function has two values, then it must also have every value in between the two.
That's so obvious (assuming the function is defined on R). We've done this as a deduction of continuity, not as a separate theorem.
 
  • #4
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These values have a pattern...
2 3 3 2
6 2 2 6
 
  • #5
FactChecker
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At the begining of the solution, it says "If A(t) ..." . So they are still defining the problem. I think that they have just put the section labels in the wrong place.

To say that the intermediate value theorem is just a "deduction of continuity" is a little too casual for some people in formal mathematical proofs. A pure mathematician would be more comfortable with this https://en.wikipedia.org/wiki/Intermediate_value_theorem#Proof . And once that theorem is established formally, they would refer to it by name. Pure math is full of casual assumptions that turned out to be wrong (thanks, Georg Cantor, you SOB).
 
  • #6
236
21
At the begining of the solution, it says "If A(t) ..." . So they are still defining the problem. I think that they have just put the section labels in the wrong place.

To say that the intermediate value theorem is just a "deduction of continuity" is a little too casual for some people in formal mathematical proofs. A pure mathematician would be more comfortable with this https://en.wikipedia.org/wiki/Intermediate_value_theorem#Proof . And once that theorem is established formally, they would refer to it by name. Pure math is full of casual assumptions that turned out to be wrong (thanks, Georg Cantor, you SOB).
Ok professor:wink:
 

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