Prove by rector method a/(sin A) = b/(sin B) = c/(sin c)

To prove the sine rule, which states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

using vector methods, let's consider a triangle \(ABC\) with vectors representing its sides. Let the vertices \(A\), \(B\), and \(C\) be represented by position vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\), respectively.

### Step-by-Step Proof Using Vectors

1. **Represent the Sides Using Vectors:**

   Consider the triangle \(ABC\), where:
   - \(\vec{AB} = \vec{c}\)
   - \(\vec{BC} = \vec{a}\)
   - \(\vec{CA} = \vec{b}\)

   Here, \(c = |\vec{c}|\), \(a = |\vec{a}|\), and \(b = |\vec{b}|\) are the magnitudes of the sides opposite to vertices \(C\), \(A\), and \(B\), respectively.

2. **Area of the Triangle Using Vectors:**

   The area of the triangle \(ABC\) can be calculated using the cross product of two sides:

   \[
   \text{Area} = \frac{1}{2} |\vec{AB} \times \vec{AC}|
   \]

   Since \(\vec{AB} = \vec{c}\) and \(\vec{AC} = -\vec{b}\), the area can also be expressed as:

   \[
   \text{Area} = \frac{1}{2} |\vec{c} \times (-\vec{b})| = \frac{1}{2} |\vec{c} \times \vec{b}|
   \]

3. **Relate the Area to the Sine of the Angle:**

   The magnitude of the cross product \(\vec{c} \times \vec{b}\) is given by:

   \[
   |\vec{c} \times \vec{b}| = |\vec{c}| |\vec{b}| \sin A
   \]

   Thus, the area can be written as:

   \[
   \text{Area} = \frac{1}{2} bc \sin A
   \]

   Similarly, using different sides of the triangle, we have:

   \[
   \text{Area} = \frac{1}{2} ca \sin B = \frac{1}{2} ab \sin C
   \]

4. **Equating the Expressions for the Area:**

   Since all three expressions represent the same area, we can equate them:

   \[
   \frac{1}{2} bc \sin A = \frac{1}{2} ca \sin B = \frac{1}{2} ab \sin C
   \]

   Simplifying by multiplying through by 2 and dividing by \(abc\), we get:

   \[
   \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
   \]

   Taking the reciprocal, we obtain the sine rule:

   \[
   \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
   \]

### Conclusion

Thus, we have proved the sine rule using vector methods.

Would you like more details on any step or further explanation? Here are some related questions:

1. How is the cross product of two vectors calculated?
2. Can we use this vector method to prove other trigonometric identities?
3. How does the sine rule apply in non-Euclidean geometries?
4. What are practical applications of the sine rule in physics and engineering?
5. Can you explain the cosine rule using vectors?

**Tip:** When proving geometric identities, visualizing the problem with diagrams can be very helpful.

Explore how to prove the sine rule \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\) using vector methods in trigonometry. This detailed step-by-step solution demonstrates the application of vector cross products and triangle area calculations. Suitable for advanced high school and college students studying trigonometry.

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