Given that $\vec{a} \cdot \vec{b} = 0$ and $\vec{a} \times \vec{b} = \vec{0}$. What can you conclude about the vectors $\vec{a}$ and $\vec{b}$?
(A) Given $\vec{a} \cdot \vec{b} = 0$. This implies that either $|\vec{a}| = 0$ or $|\vec{b}| = 0$,or $\vec{a} \perp \vec{b}$ (if both are non-zero vectors).
Given $\vec{a} \times \vec{b} = \vec{0}$. This implies that either $|\vec{a}| = 0$ or $|\vec{b}| = 0$,or $\vec{a} \parallel \vec{b}$ (if both are non-zero vectors).
Since two non-zero vectors cannot be both perpendicular and parallel to each other simultaneously,the only possibility is that at least one of the vectors must be a zero vector.
Therefore,we conclude that either $\vec{a} = \vec{0}$ or $\vec{b} = \vec{0}$.
Given $\vec{a} \times \vec{b} = \vec{0}$. This implies that either $|\vec{a}| = 0$ or $|\vec{b}| = 0$,or $\vec{a} \parallel \vec{b}$ (if both are non-zero vectors).
Since two non-zero vectors cannot be both perpendicular and parallel to each other simultaneously,the only possibility is that at least one of the vectors must be a zero vector.
Therefore,we conclude that either $\vec{a} = \vec{0}$ or $\vec{b} = \vec{0}$.
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