1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Show orthogonality of vector-valued functions

  1. Oct 9, 2016 #1
    I have this exercise on my book and I believe it is quite simple to solve, but I'm not sure if I did good, so here it is

    1. The problem statement, all variables and given/known data
    given a vector B ∈ ℝn, B ≠ 0 and a function F : ℝ → ℝn such that F(t) ⋅ B = t ∀t and the angle φ between F'(t) and B is constant with respect to t, show that F''(t) is perpendicular to F'(t);

    2. Relevant equations
    perpendicular vectors: F''(t) ⋅ F'(t) = 0

    3. The attempt at a solution
    Since I have to show that the scalar product between F'' and F' is 0, i would obviously try to derive the given eqs and find a relation between the two derivatives
    F(t) ⋅ B = t ⇒ d/dt(F(t) ⋅ B) = d/dt(t) ⇒ F'(t) ⋅ B = 1
    F'(t) ⋅ B = |F'(t)B|cosφ = 1 ⇒ d/dt(|F'(t)B|cosφ) = d/dt(1) ⇒ |F''(t)B|cosφ = 0
    cosφ ≠ 0 since F'Bcosφ = 1, and B ≠ 0, then it must be F''(t) = 0 ⇒ F''(t) ⋅ F'(t) = 0

    Is this correct? Can I say that the zero vector is perpendicular to F' or did I miss something?
  2. jcsd
  3. Oct 9, 2016 #2


    User Avatar
    Homework Helper

    It is not generally true that [itex]\|F'\|' = \|F''\|[/itex]. On differentiating [itex]\|F'\|\|B\|\cos \phi[/itex] with respect to [itex]t[/itex] you should obtain [tex]
    \|B\|\cos \phi\frac{d\|F'\|}{dt} = 0[/tex] since [itex]\frac{d}{dt}\|B\| = \frac{d\phi}{dt} = 0[/itex].

    To proceed further you will need the identity [tex]
    \frac{d}{dt}\|F'\|^2 = \frac{d}{dt} (F' \cdot F') = 2F' \cdot F''.[/tex]
  4. Oct 9, 2016 #3
    Oh I see that, I should have been more careful. Thank you very much.
  5. Oct 9, 2016 #4


    Staff: Mentor

    @mastrofoffi, please post questions of this nature in the Calculus & Beyond section, not the Precalculus section. If the question involves derivatives, it should NOT go in the Precalc section.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted