How is the Law of Sines Derived Using the Cross Product?

[tronter]

How would you use the cross product to derive the law of sines?

$$A \times B = |A||B| \sin \theta$$.

Law of sines: $$\frac{\sin A}{A} = \frac{\sin B}{b} = \frac{\sin C}{c}$$.

The cross product gives the area of the parallelogram formed by the vectors.

 

[wildman]

This looks like a homework problem. Next time, ask your homework questions in the homework section. You will get a lot quicker answer if you do.

Having said that, you need to show some work -- we don't just give answers here... But here are some clues:

What does having the cross products of each set of four vectors equal to the area of the parallelogram say about how the cross products relate? Are they the same, different? Draw a picture, set up some equations and play around a bit.

Do the above and then if you still don't get it, show us what you have done and we can give you some more clues...

 

[tronter]

The cross products are equal to each other. And I think its 3 cross products.

So $$A \times B = A \times C = B \times C$$

or $$|A||B| \sin C = |A||C| \sin B = |B||C| \sin A$$.

Hence $$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$

 

[uart]

Yep that's it. All you have to do now is divide throughout by the product (|a| |b| |c|).