Law of Sines Proof w/ Vector Calculus

In summary, The Law of Sines can be proven using vector methods by considering the cross-product of vectors a, b, and c. By simplifying the equation, absin(C) = bcsin(A) = casin(B), we can arrive at the familiar form of the law of sines. This method provides a geometric approach to understanding the relationship between the angles and sides of a triangle, and can be a helpful tool in solving related problems.
  • #1
jimmythegent
4
0
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.
 
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  • #2
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?
 
  • #3
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.
 

Related to Law of Sines Proof w/ Vector Calculus

1. What is the Law of Sines?

The Law of Sines is a mathematical relationship that describes the relationship between the angles and sides of a triangle. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles of a triangle.

2. How does Vector Calculus relate to the Law of Sines?

Vector Calculus is a branch of mathematics that deals with the study of vector fields, which are functions that assign a vector to each point in space. The Law of Sines can be derived using vector calculus by considering the position vectors of the vertices of a triangle and using properties of vector operations such as cross product and dot product.

3. What is the proof for the Law of Sines using Vector Calculus?

The proof for the Law of Sines using Vector Calculus involves setting up a system of equations using the position vectors of the triangle and solving for the length of each side using vector operations. By equating these lengths to the sine of their respective opposite angles, the constant ratio of the Law of Sines is obtained.

4. What are the benefits of using Vector Calculus to prove the Law of Sines?

Using Vector Calculus to prove the Law of Sines provides a more geometric and intuitive understanding of the relationship between the angles and sides of a triangle. It also allows for a more general proof that can be applied to non-right triangles, whereas the traditional proof of the Law of Sines only applies to right triangles.

5. Are there any limitations to using Vector Calculus to prove the Law of Sines?

One limitation is that Vector Calculus may be more difficult for some students to understand compared to the traditional proof of the Law of Sines. Additionally, the proof using Vector Calculus requires a solid understanding of vector operations and may not be accessible to those without a background in mathematics.

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