• Support PF! Buy your school textbooks, materials and every day products Here!

Parallelism of Time-varying Vectors

  • #1
232
20

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...

 

Attachments

Answers and Replies

  • #2
FactChecker
Science Advisor
Gold Member
5,522
2,018
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.
 
  • Like
Likes baldbrain
  • #3
232
20
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
232
20
These values have a pattern...
2 3 3 2
6 2 2 6
 
  • #5
FactChecker
Science Advisor
Gold Member
5,522
2,018
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
232
20
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:
 

Related Threads on Parallelism of Time-varying Vectors

  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
13
Views
2K
  • Last Post
Replies
4
Views
8K
  • Last Post
Replies
17
Views
3K
  • Last Post
Replies
5
Views
15K
  • Last Post
Replies
6
Views
579
  • Last Post
Replies
3
Views
3K
Replies
7
Views
20K
  • Last Post
Replies
3
Views
3K
Replies
6
Views
6K
Top