1. **State the problem:** We want to prove that three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ are coplanar, meaning they lie in the same plane. 2. **Formula and rule:** Three vectors are coplanar if the scalar triple product is zero. The scalar triple product is given by: $$\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$$ This means the volume of the parallelepiped formed by the three vectors is zero, indicating they lie in the same plane. 3. **Explanation:** - Compute the cross product $\vec{b} \times \vec{c}$. - Then compute the dot product of $\vec{a}$ with the result. - If the result is zero, the vectors are coplanar. 4. **Intermediate work example:** Suppose $\vec{a} = (a_1, a_2, a_3)$, $\vec{b} = (b_1, b_2, b_3)$, and $\vec{c} = (c_1, c_2, c_3)$. Calculate: $$\vec{b} \times \vec{c} = (b_2 c_3 - b_3 c_2, b_3 c_1 - b_1 c_3, b_1 c_2 - b_2 c_1)$$ Then: $$\vec{a} \cdot (\vec{b} \times \vec{c}) = a_1 (b_2 c_3 - b_3 c_2) + a_2 (b_3 c_1 - b_1 c_3) + a_3 (b_1 c_2 - b_2 c_1)$$ If this equals zero, the vectors are coplanar. 5. **Summary:** To prove three vectors are coplanar, calculate the scalar triple product. If it equals zero, the vectors lie in the same plane.