Question

If \[\vec{a} \text{ and }  \vec{b}\] are two vectors of magnitudes 3 and \[\frac{\sqrt{2}}{3}\]  espectively such that \[\vec{a} \times \vec{b}\] is a unit vector. Write the angle between \[\vec{a} \text{ and }  \vec{b} .\]

 

 

 

 

Answer 1

\[\text{ Let } \theta \text{ be the angle between } \vec{a} \text{ and }  \vec{b} .\]
\[\text{ It is given that } \vec{a} \times \vec{b} \text{ is a unit vector } .\]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = 1\]
\[\text{ We know } \]
\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta\]
\[ \Rightarrow 1 = \left( 3 \right) \left( \frac{\sqrt{2}}{3} \right) \sin \theta\]
\[ \Rightarrow \sin \theta = \frac{1}{\sqrt{2}}\]
\[ \Rightarrow \theta = {45}^o , {135}^o\]