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Problem 3


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Problem

Show that if u and v are vectors then
|v| u + |u| v

bisects the angle between u and v.

First notice that we only need to show that (u+v) bisects the angle between u and v when |u|=|v|=1. The general case will follow from here, for if the result is true for vectors of length 1 then,

(u/|u|) + (v/|v|)

will bisect the angle between u/|u| and v/|v| which is the same as the angle between u and v (we are only changing their lengths). Also, if a vector b bisects an angle then so does any scalar multiple of it. Thus, multiplying by (|u|*|v|) we obtain the original result.

OK but we still need to show that u+v does the trick. That is not difficult to see. We loose no generality if we choose a coordinate system where:

i=u, and j on the plane containing u and v


> u := vector([1,0]); v := vector([cos(t),sin(t)]);


                                 u := [ 1, 0 ]

                            v := [ cos(t), sin(t) ]




The vector v forms an angle of "t" with u,


> angle(u,u+v);


                                      1 + cos(t)
                      arccos(----------------------------)
                                          2         2 1/2
                             ((1 + cos(t))  + sin(t) )




so this is the angle between u and u+v... is this t/2 ? First, simplify with trig,


> ang := simplify(",trig);


                                          1 + cos(t)
                        ang := arccos(-----------------)
                                                    1/2
                                      (2 + 2 cos(t))




we are getting there... but not yet. Now let's use the formula for cos(t) in terms of cos(t/2). Let's make s=t/2 and simplify.


> ang := subs(t=2*s,ang);


                                         1 + cos(2 s)
                       ang := arccos(-------------------)
                                                     1/2
                                     (2 + 2 cos(2 s))



> ang := arccos(expand(cos(ang)));

                                           1/2       2
                                          4    cos(s)
                        ang := arccos(1/2 ------------)
                                                 2 1/2
                                          (cos(s) )



> simplify(");

                          arccos(csgn(cos(s)) cos(s))




The inside of arccos is nothing but |cos(s)| and since we are assuming that 0 < s < Pi/2, the above is just s.
Q.E.D.


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Thu Sep 19 10:27:27 EDT 1996