Question

For any two vectors  \[\vec{a} \text{ and }  \vec{b}\] write the value of \[\left( \vec{a} . \vec{b} \right)^2 + \left| \vec{a} \times \vec{b} \right|^2\] in terms of their magnitudes.

 

 

Answer 1

\[\left( \vec{a} . \vec{b} \right)^2 + \left| \vec{a} \times \vec{b} \right|^2 \]

\[ = \left( \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta \right)^2 + \left( \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta \right)^2 \]

\[ = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \left( \cos^2 \theta + \sin^2 \theta \right)\]

\[ = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \left( 1 \right)\]

\[ = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2\]