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Find a bisecting angle

  1. Feb 1, 2015 #1
    1. The problem statement, all variables and given/known data
    If c = |a|b + |b|a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b.

    2. Relevant equations
    Angle between a & b is cos-1(a dot b)/(|a||b|)
    Angle between a & c is cos-1(a dot c)/(|a||c|)
    Angle AC is half of angle AB

    3. The attempt at a solution
    Given c = |a|b + |b|a, I plug that into the equation for the angle between a and c. I eventually get (|a||b|)(|a|b + |b|a2 = (2|a||c|)(a dot b). Is this right? I would also like to confirm:

    - if (a dot a) is always equal to |a|^2
    - how to differentiate absolute value and magnitude as they use the same symbol
    - When I got (2|a||c|)(a dot b), do I do regular multiplication or use the dot product?
  2. jcsd
  3. Feb 1, 2015 #2


    Staff: Mentor

    This doesn't look right. If θ is the angle between a and c, you should have ##cos(θ) = \frac{a \dot c}{|a||c|}##. When you substitute in c and |c|, there should be some stuff in the denominator. I don't see that in what you have.
    Absolute values apply to real numbers. For a vector, |v| means the magnitude of the vector.
    Regular multiplication, if I'm understanding what you are asking. The two quantities in parentheses are real numbers.
  4. Feb 1, 2015 #3


    Staff: Mentor

    A simpler approach than you're taking is to calculate the cosines of the two angles; i.e., the angle between a and c, and the angle between c and b. As expected, these turn out to be equal.
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