Question

Prove that \[\vec{A} . \left( \vec{A} \times \vec{B} \right) = 0\].

Answer 1

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.