Law of Sines Proof w/ Vector Calculus

[jimmythegent]

Hey guys, it's my first post here so please don't chew my head off if I do something forbidden, hahah.

Homework Statement

Prove the Law of Sines using Vector Methods.

Homework Equations

sin(A)/a = sin(B)/b = sin(C)/c

The Attempt at a Solution

Since axb=sin(C), I decided to try getting the cross product and then trying to match it to the equation. I gave each of the points coordinates and I decided to come here for help when the problem became gigantic and hardly manageable. I've spent a couple of hours on this and I'm not getting anywhere.

If you've got any suggestions for a simpler solution or even just a nudge in the right direction, I'd be very grateful.

 

[robphy]

Here's a hint:
Start by drawing a triangle with vectors, a, b, c.
Can you apply the cross-product with any pair of these vectors?

 

[jimmythegent]

Yay, got it! Thanks!

I said
axb = area of parallelogram
bxc = area of parallelogram
cxa = area of parallelogram

so axb = bxc = cxa
then absin(C) = bcsin(A) = casin(B)

then crossed stuff out and got the law of sines.