1. **Problem 1:** Find the geometrical interpretation of the cross product of two vectors and prove the sine rule in triangle ABC using vector methods. 2. **Geometrical interpretation of cross product:** The cross product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\), denoted \(\mathbf{a} \times \mathbf{b}\), is a vector perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). Its magnitude equals the area of the parallelogram formed by \(\mathbf{a}\) and \(\mathbf{b}\), given by: $$|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin \theta$$ where \(\theta\) is the angle between \(\mathbf{a}\) and \(\mathbf{b}\). 3. **Proof of sine rule using vectors:** Consider triangle ABC with sides \(a = |\mathbf{BC}|\), \(b = |\mathbf{AC}|\), and \(c = |\mathbf{AB}|\). Let \(\mathbf{AB} = \mathbf{c}\), \(\mathbf{BC} = \mathbf{a}\), and \(\mathbf{CA} = \mathbf{b}\). The area of triangle ABC can be expressed as half the magnitude of the cross product of two side vectors: $$\text{Area} = \frac{1}{2} |\mathbf{AB} \times \mathbf{AC}| = \frac{1}{2} |\mathbf{c} \times (-\mathbf{b})| = \frac{1}{2} |\mathbf{b} \times \mathbf{c}|$$ Similarly, the area can be written using other pairs: $$\text{Area} = \frac{1}{2} |\mathbf{a} \times \mathbf{b}| = \frac{1}{2} |\mathbf{b} \times \mathbf{c}| = \frac{1}{2} |\mathbf{c} \times \mathbf{a}|$$ 4. Using the magnitude of cross product: $$\text{Area} = \frac{1}{2} ab \sin C = \frac{1}{2} bc \sin A = \frac{1}{2} ca \sin B$$ 5. Equate these expressions: $$ab \sin C = bc \sin A = ca \sin B$$ Divide each term by \(abc\): $$\frac{\sin C}{c} = \frac{\sin A}{a} = \frac{\sin B}{b}$$ This is the sine rule: $$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$ --- 6. **Problem 2:** Solve the system of equations using Gauss-Seidel method and find the error. (Since the system of equations is not provided, please provide the system to proceed.)