# Deriving law of sines from cross product

• B
• Mr Davis 97
In summary, the conversation discusses trying to derive the law of sines from the cross product. The method involves taking the cross product of one vector with the sum of the other two vectors and using the resulting equations to compare the ratios of the opposite angles with the respective lengths of the sides. This allows for the derivation of the third angle's ratio in terms of the lengths of the sides.
Mr Davis 97
I am trying to derive the law of signs from the cross product.

First, we have three vectors ##\vec{A} ~\vec{B} ~\vec{C}## such that ##\vec{A} + \vec{B} + \vec{C} = 0##. This creates a triangle. Then, we label the angles opposite the respective sides as a, b, and c. I am not sure where to go from here... We could take the cross product of each combination of ##\vec{A}## and ##\vec{B}##, but these cross products aren't necessarily equal, so can't set them equal to derive the law of sines... Any help would be appreciated.

Mr Davis 97 said:
I am trying to derive the law of signs from the cross product.

First, we have three vectors ##\vec{A} ~\vec{B} ~\vec{C}## such that ##\vec{A} + \vec{B} + \vec{C} = 0##. This creates a triangle. Then, we label the angles opposite the respective sides as a, b, and c. I am not sure where to go from here... We could take the cross product of each combination of ##\vec{A}## and ##\vec{B}##, but these cross products aren't necessarily equal, so can't set them equal to derive the law of sines... Any help would be appreciated.
If you take ## C \times (A+B+C)=0 ## and ## C \times C=0 ## Then ## |A||C|sin(\theta_1)=|B||C|sin(\theta_2) ## (neglecting a minus sign which is of little significance).Result is ## sin(\theta_1)/|B|=sin(\theta_2)/|A| ##

How do I get ##sin(\theta_3)/|C|## in there? That was my main problem.

Mr Davis 97 said:
How do I get ##sin(\theta_3)/|C|## in there? That was my main problem.
That will show up if you take ## B \times (A+B+C)=0 ## (or ## A \times (A+B+C)=0 ##). You can only do two angles at a time by this method. Comparing two of the equations will tie the 3rd one in there. e.g. ## B \times ## gives you ## sin(\theta_2)/|A|=sin(\theta_3)/|C| ##.

Awesome. Thanks! While the way you put it is very understandable, but I'm not really sure how you came to the conclusion that the cross product of one of the vectors with the sum of the other three would lead you to the proof... Basically, "how would I have thought of that?"

## 1. What is the law of sines?

The law of sines is a mathematical principle that relates the sides and angles of a triangle. It states that the ratio of each side of a triangle to the sine of its opposite angle is always equal.

## 2. How is the law of sines derived from cross product?

The law of sines can be derived from the cross product of two vectors in the plane of the triangle. By setting up an equation using the sine of the angles and the cross product of the vectors, we can solve for the ratio of the sides and ultimately derive the law of sines.

## 3. What are the benefits of using the cross product to derive the law of sines?

Using the cross product to derive the law of sines provides a geometric interpretation of the law, making it easier to understand and visualize. It also allows for a more general approach, as the cross product can be applied to any triangle, not just right triangles.

## 4. Are there any limitations to deriving the law of sines from cross product?

One limitation is that the cross product method can only be applied to triangles in the plane. It also requires some knowledge of vector operations and may not be as intuitive for those unfamiliar with vectors.

## 5. How is the law of sines used in real-world applications?

The law of sines is commonly used in fields such as engineering, physics, and navigation. It can be used to solve problems involving triangles, such as finding the distance between two points or determining the height of a building. It is also used in surveying and astronomy to calculate distances and angles.

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