Parallelism of Time-varying Vectors

[baldbrain]

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.

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...

 

[FactChecker]

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.

 

[baldbrain]

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.

 

[baldbrain]

These values have a pattern...
2 3 3 2
6 2 2 6

 

[FactChecker]

At the beginning 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).

 

[baldbrain]

At the beginning 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