1. Not finding help here? Sign up for a free 30min 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!

Vector Proof

  1. Jun 21, 2012 #1
    1. The problem statement, all variables and given/known data

    Given vector a and b, let vector v = (|a|b + |b|a) / (|a| + |b|). Show that vector v bisects the angle created by vector b and a.

    2. Relevant equations

    cos(θ) = a dot b / (|a||b|)


    3. The attempt at a solution

    I used the dot product equation to find angle between a and v.

    cosθ = (a dot v) / |a||v|

    i substituted ((|a|b + |b|a) / (|a| + |b|)) for v.

    cosθ = (a dot ((|a|b + |b|a) / (|a| + |b|)) / |a||v|

    I distributed the (a dot ((|a|b + |b|a)

    cosθ ( ((|a|b dot a) + (|b| |a|^2) ) / (|a| + |b|)) / (|a||b|)

    I then tried to get the cosθ of b dot w to the same thing. I tried to get the two angles equal to each other. My idea is that if the angles formed by a and v and b and v are equal then v bisects the two angles.
     
  2. jcsd
  3. Jun 21, 2012 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    So you have
    $$\cos\theta_{av} = \frac{a(\vec{a}\cdot\vec{b})+a^2b}{(a+b)av}$$ where ##a=|\vec{a}|##, ##b=|\vec{b}|##, and ##v=|\vec{v}|##, and the corresponding expression for ##\cos\theta_{bv}##. So far so good. Try calculating
    $$\frac{\cos\theta_{av}}{\cos\theta_{bv}}$$ and show it's equal to 1.
     
  4. Jun 21, 2012 #3
    Meson,

    The problem becomes easy when you realize that if you add two different vectors of equal lengths, then you get a third vector whose direction bisects the angle of the first two vectors. Try it and see. That takes care of the geometry. Now see if you can figure out the algebra.

    Ratch
     
    Last edited: Jun 22, 2012
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Vector Proof
  1. Vector proof (Replies: 10)

  2. Vector Proof (Replies: 1)

  3. Vector Proof (Replies: 2)

  4. Vector Proof (Replies: 0)

  5. Vector Proof (Replies: 11)

Loading...