How to Prove the Sine Rule Using Cross Product?

[iasc]

Could anyone tell me how to use the cross product to prove the sine rule

 

[rl.bhat]

Area of a triangle of side a.b and c is
A = 1/2*axb = 1/2absinC
Similarly 1/2*bxc = 1/2 bcsinA and so on
So
absinC = bcsinA = casinB. Dividing abc to all we get
sinA/a = sinB/b = sinC/c

 

[luisfigo777]

Here AB,BC,CA ,a,b,c are vectors and AB=a BC=b CA=c
in a triangle ABC,
AB + BC + CA = 0
a + b + c= 0
a x b =b x c = c x a(proved using above statement just take b and c to other side and take crossproduct with b on both sides first and then with c)
la x bl= |b x c|= |c x a |
|a||b| SinC= |b||c|SinA=|a||c| SinB
dividing by |a||b||c|
we get Sine formula