Vector

Let
→
OA be A = 12i + 5j ,
→
OB be B = 3i - 4j ,
→
OP be P = xi + yj , is the angle bisector of A and B.then
cos∠AOP
= A * P / ( |A| |P| )
= (12x + 5y) / ( √(12² + 5²) √(x² + y²) )
= (12x + 5y) / ( 13√(x² + y²) )cos∠BOP
= B * P / ( |B| |P| )
= (3x - 4y) / ( √(3² + 4²) √(x² + y²) )
= (3x - 4y) / ( 5√(x² + y²) )
∵ P is the angle bisector of A and B.
∴ cos∠AOP = cos∠BOP (12x + 5y) / ( 13√(x² + y²) ) = (3x - 4y) / ( 5√(x² + y²) )
60x + 25y = 39x - 52y
21x + 77y = 0
y = - 3x / 11 ... (1)
For the unit vector, |P| = √(x² + y²) = 1 ... (2)Substitute (1) into (2) :
√(x² + (- 3x/11)²) = 1
(1 + 9/121)x² = 1
x² = 121/130
x = 11√130 / 130 or - 11√130 / 130 (rejected)
y = - 3/11 * 11√130 / 130 = - 3√130 / 130
The unit vector of the angle bisector of A and B is
(11√130 / 130) i - (3√130 / 130) j