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Question
Prove that \[\vec{A} . \left( \vec{A} \times \vec{B} \right) = 0\].
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Solution
To prove:
\[\vec{A} . \left( \vec{A} \times \vec{B} \right) = 0\]
Proof: Vector product is given by \[\vec{A} \times \vec{B} = \left| \vec{A} \right|\left| \vec{B} \right| \sin \hat {n}\]
\[\left| \vec{A} \right|\left| \vec{B} \right| \sin\hat { n }\]
is a vector which is perpendicular to the plane containing
\[\vec{A} \text { and } \vec{B}\] This implies that it is also perpendicular to \[\vec{A}\]. We know that the dot product of two perpendicular vectors is zero.
∴ \[\vec{A} . \left( \vec{A} \times \vec{B} \right) = 0\]
Hence, proved.
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