Question

If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then write the value of \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 .\]

 

 

Answer 1

\[\text{ It is given that } \vec{a} \text{ and } \vec{b} \text{ are unit vectors } .\]
\[ \Rightarrow \left| \vec{a} \right| = \left| \vec{b} \right| = 1 . . . (1)\]
\[\text{ Now } , \]
\[ \left( \vec{a} . \vec{b} \right)^2 + \left| \vec{a} \times \vec{b} \right|^2 \]
\[ = \left( \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta \right)^2 + \left( \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta \right)^2 \]
\[ = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \left( \cos^2 \theta + \sin^2 \theta \right)\]
\[ = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \left( 1 \right)\]
\[ = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \]
\[ = 1^2 1^2 [\text{ From }  (1)]\]

= 1