Question
Prove that $\vec{\text{A}}.(\vec{\text{A}}\times\vec{\text{B}})=0.$
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Answer
$\vec{\text{A}}.(\vec{\text{A}}\times\vec{\text{B}})=0$ (claim)As, $\vec{\text{A}}\times\vec{\text{B}}=\text{AB}\sin\theta \ \hat{\text{n}}$
$\text{AB}\sin\theta \ \hat{\text{n}}$ is a vector which is perpendicular to the plane containing $\vec{\text{A}}$ and $\vec{\text{B}},$ this implies that it is also perpendicular to $\vec{\text{A}}.$ As dot product of two perpendicular vector is zero.
Thus $\vec{\text{A}}.(\vec{\text{A}}\times\vec{\text{B}})=0.$
$\text{AB}\sin\theta \ \hat{\text{n}}$ is a vector which is perpendicular to the plane containing $\vec{\text{A}}$ and $\vec{\text{B}},$ this implies that it is also perpendicular to $\vec{\text{A}}.$ As dot product of two perpendicular vector is zero.
Thus $\vec{\text{A}}.(\vec{\text{A}}\times\vec{\text{B}})=0.$
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