Proving Angle Bisector of A and B with Vector Magnitudes

[thenewbosco]

Prove that $$\frac{|B|A+|A|B}{|A|+|B|}$$ is the bisector of the angle formed by A and B. where i have used normal text for vector and abs value bars to represent magnitude of vector.

i have no clue how to get started on this. i have tried many approaches such as constructing a triangle with a, b, and b-a, but i cannot seem to make any progress. a couple of hints on getting started would be appreciated

 

[courtrigrad]

Write two equations:

$$ax+by+c = 0$$
$$cx+dy+e = 0$$

 

[thenewbosco]

and what do these equations represent?

 

[Office_Shredder]

I don't know off the top of my head (i'm not a vector geometry expert), but if you call the bisector vector C, taking $$A \cdot C$$ and $$B \cdot C$$ and knowing the cosine half angle formula should be a decent way to start

 

[Hurkyl]

a couple of hints on getting started would be appreciated

You want to know that the vector you constructed (I'll call it C) is the angle bisector of A and B. Therefore, you want to know:

(1) The angle between A and B
(2) The angle between A and C
(3) The angle between B and C

don't you?

 

[courtrigrad]

Suppose that A and B intersect at some point Q, and R is some point on A , and S is some point on B . Write the vector equations of the individual lines, and then of the bisector.

 

[thenewbosco]

so i wrote A=Q+tQR
and B=Q+tQS,
as my two vector equations...how can i write the bisector