This applet shows you a triangle (created by adding 2 vectors together) and allows you to drag the vertices around. The text surrounding the triangle gives a vector-based proof of the Law of Sines.
As you drag the vertices (vectors) the magnitude of the cross product of the 2 vectors is updated.
The proof shows that any 2 of the 3 vectors comprising the triangle have the same cross product as any other 2 vectors.
Hand-wavy proof:
This makes sense because the cross product of any 2 gives the Area of the parallelogram which can be formed.
The Area of the triangle must be half that of the parallelogram (regardless of which 2 vectors were chosen, so the Area of the parallelogram will be the same so the cross-products must be the same QED).)

Can you confirm the Law of Sines numerically starting from what I've provided?