For two vectors, can A×B=0 and A.B=0 simultaneously act?

OR: If A and B are non zero vectors, is it possible for A×B and A.B both to be zero?

For the two vectors, \[\vec{A}×\vec{B}=AB\sin\theta\;\hat{n}\] \[0=AB\sin\theta\;\hat{n}\] \[\implies \text{either}\;A=0\;\text{or}\;B=0\;\text{or}\;\sin\theta=0\]

Also, \[\vec{A}×\vec{B}=AB\cos\theta\] \[0=AB\cos\theta\] \[\implies \text{either}\;A=0\;\text{or}\;B=0\;\text{or}\;\cos\theta=0\]

If $\vec{A}$ and $\vec{B}$ are non zero vectors, then $\sin\theta$ and $\cos\theta$ should be zero simultaneously which is impossible. So, in this situation, $\vec{A}×\vec{B}=0$ and $\vec{A}.\vec{B}=0$ cannot act simultaneously.

Therefore, for $\vec{A}×\vec{B}$ and $\vec{A}.\vec{B}$ both to be zero, $\vec{A}$ and $\vec{B}$ must be null vectors.

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