Proof of law of sines using cross product

[BelieverOf143___]

Homework Statement

Prove the law of sines using the vector cross product

Relevant Equations

Is there some relation between the vector cross product and sin?

Can somebody please give me a hint for this problem?

 

[berkeman]

Homework Statement: Prove the law of sines using the vector cross product
Relevant Equations: Is there some relation between the vector cross product and sin?

Can somebody please give me a hint for this problem?

What problem? What do you know about how to compute cross products?

 

[BelieverOf143___]

Well for the cross product I found a general way on how to evaluate the cross product for two dimensions (a general formula for the value of the z-coordinate)

 

[BelieverOf143___]

I know how to compute them

 

[haruspex]

When given a task like this, you have to start with the definitions you are expected to work from. State the law of sines and the definition of the cross product you've been given.
If the latter is in terms of its magnitude being $|\vec a||\vec b|\sin(\theta)$, try combining that with representing a triangle as a summation of two vectors.

 

[wrobel]

Nice way to prove the law of sines but what about to get $2R$ in the same spirit? :)

 

[martinbn]

Can somebody please give me a hint for this problem?

Consider the sides of the triangle as vectors. Take one of them. It is the sum of the other two. Take the cross product of that side by itself.

 

[martinbn]

Nice way to prove the law of sines but what about to get $2R$ in the same spirit? :)

With the standard notations, let $D$ be the other end of the diameter from $A$, then $\overrightarrow{DB}=\overrightarrow{DA}+\overrightarrow{AB}$. Take cross product with $\overrightarrow{DB}$, and use that the angle at $D$ is equal to the one at $C$.

 

[wrobel]

With the standard notations, let D be the other end of the diameter from A, then DB→=DA→+AB→. Take cross product with DB→, and use that the angle at D is equal to the one at C.

this is essentially the same standard proof that everybody knows

 

[Herman Trivilino]

Is there some relation between the vector cross product and sin?

Not sine, the Law of Sines. For any triangle the length of a side is proportional to the sine of the angle opposite to it.

 

[haruspex]

Not sine, the Law of Sines. For any triangle the length of a side is proportional to the sine of the angle opposite to it.

I see no reason to doubt @BelieverOf143___ understands that. If the vector product rule can be used to prove the law of sines then there must be some connection between the vector product and the sine function.
The OP's question indicates ignorance of such a connection. So to assist, we need to know what definition of the cross product is to be used.