Oh i just wanted to reach the conclusion with the last paragraph quickly. I constructed that vector as an arbitrary vector that will act as a bisector. Didnt want to use the fact that its half a rhombus and in essence that is the way the complex method I was referring to would've been established. The other reason I did not put it as the form you referred to above was because i just wished to prove that the vector c was perpendicular. Didnt need any other stuff for it so didn't go with the definition you have provided above; so the proof
should be sufficient to say the very least.
It is possible to simply skip defining
explicitly, and even if I were to do so, I would not offer an ambiguous definition of
as you have done. It simply invites questions as to why that definition was chosen and whether the
that you have defined actually has the properties that you claim. In this situation, it is odd to define
by means of a dot product as it means that
is not a unique vector as there is more than one
that satisfies your definition. Under your definition,
must satisfy
and have an angle between it and
as
- but there are two vectors that fit these criteria and only one of them is the bisector that you seek.
It is sufficient to provide a limited defition: that
is a vector that bisects the angles between
and
, so that the angle between
and
and the angle between
and
are the same (i.e.
) and thus that the angle between
and
is
subject to the requirements that
and
.
In other words, by simply naming
and defining it as having the properties of a bisector, your proof that the dot product
follows quickly and without distraction. As a marker, I would have to stop and consider whether your
is actually valid, and the definition itself is not used for the proof in any event so it isn't actually needed.