Question
Can you have $\overrightarrow{\text{A}}\times\overrightarrow{\text{B}}=\overrightarrow{\text{A}}\cdot\overrightarrow{\text{B}}$ with $\text{A}\neq0$ and $\text{B}\neq0?$ What if one of the two vectors is zero?
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Answer
Never, because $\overrightarrow{\text{A}}\times\overrightarrow{\text{B}}=\overrightarrow{\text{A}}\cdot\overrightarrow{\text{B}}$ is a vector quantity and $\overrightarrow{\text{A}}.\overrightarrow{\text{B}}$ is a scalar quantity and a vector can never be equated to a scalar, even if their magnitudes are zero. So even if one of the two vectors is zero $\overrightarrow{\text{A}}\times\overrightarrow{\text{B}}$ gives a null vector and $\overrightarrow{\text{A}}.\overrightarrow{\text{B}}$ is simply zero.
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