1. **Problem statement:** We want to generally prove and explain the orthogonality of vectors using the scalar product (dot product) and vector product (cross product). 2. **Definitions and formulas:** - The scalar product of two vectors $\vec{a} = (a_1,a_2,a_3)$ and $\vec{b} = (b_1,b_2,b_3)$ is defined as: $$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$$ - The vector product (cross product) is defined as: $$\vec{a} \times \vec{b} = \begin{pmatrix} a_2b_3 - a_3b_2 \\ a_3b_1 - a_1b_3 \\ a_1b_2 - a_2b_1 \end{pmatrix}$$ 3. **Orthogonality and scalar product:** - Two vectors are orthogonal if their scalar product is zero: $$\vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = 0$$ - This means the cosine of the angle between them is zero, i.e., the angle is $90^\circ$. 4. **Orthogonality and vector product:** - The vector product $\vec{a} \times \vec{b}$ produces a vector perpendicular to both $\vec{a}$ and $\vec{b}$. - Thus, the vector product is useful to find a vector orthogonal to two given vectors. 5. **Summary:** - The scalar product tests if two vectors are orthogonal by checking if $\vec{a} \cdot \vec{b} = 0$. - The vector product generates a vector orthogonal to the plane spanned by $\vec{a}$ and $\vec{b}$. **Final answer:** Orthogonality of vectors can be checked by the scalar product being zero, and the vector product yields a vector orthogonal to both original vectors.