△ABC中,向量AB与向量AC的内积=1,向量AB与向量BC的内积=—31.求AB边的长度.2.求sin(A-B)/sinC的值.

Question

△ABC中,向量AB与向量AC的内积=1,向量AB与向量BC的内积=—3
1.求AB边的长度.
2.求sin(A-B)/sinC的值.

Answer 1

1.
AB*AC=1 (1)
AB*BC=-3 (2)
(1)-(2)
=> AB*(AC-BC)=4
=> AB*AB=4
=> |AB|=2
2.
sin(A-B)/sinC
=sin(A-B)/sin(A+B)
=(sinAcosB-cosAsinB)/(sinAcosB-cosAsinB)
=(tanA-tanB)/(tanA+tanB) (*)
向量AB与向量AC的内积=1,向量AB与向量BC的内积=—3
=>
cbcosA=1,cacosB=-3
=>cosA/cosB=-1/3 (3)
therefore sinA/sinB=a/b (4)
(4)/(3)
=>tanA/tanB=-3/1
=>tanA=-3tanB
=>(*)=2
so sin(A-B)/sinC=2