1. **Problem statement:** Given that the cross product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is zero, i.e., $\mathbf{u} \times \mathbf{v} = \mathbf{0}$, what can we deduce about $\mathbf{u}$ and $\mathbf{v}$? 2. **Recall the formula and properties:** The cross product $\mathbf{u} \times \mathbf{v}$ is a vector perpendicular to both $\mathbf{u}$ and $\mathbf{v}$, with magnitude given by: $$ |\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin \theta $$ where $\theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$. 3. **Important rule:** The cross product is zero if and only if the vectors are parallel or one of the vectors is the zero vector. 4. **Deduction:** Since $\mathbf{u} \times \mathbf{v} = \mathbf{0}$, it implies: $$ |\mathbf{u}| |\mathbf{v}| \sin \theta = 0 $$ 5. **Analyze the equation:** This product is zero if any of the following is true: - $|\mathbf{u}| = 0$ (i.e., $\mathbf{u} = \mathbf{0}$) - $|\mathbf{v}| = 0$ (i.e., $\mathbf{v} = \mathbf{0}$) - $\sin \theta = 0$ which means $\theta = 0$ or $\theta = \pi$, so $\mathbf{u}$ and $\mathbf{v}$ are parallel (pointing in the same or opposite directions). 6. **Conclusion:** Therefore, if $\mathbf{u} \times \mathbf{v} = \mathbf{0}$, then $\mathbf{u}$ and $\mathbf{v}$ are either parallel vectors or one (or both) of them is the zero vector.