Question

If the vectors \[\vec{a}\] and \[\vec{b}\]  are such that \[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = \frac{2}{3}\] and \[\vec{a} \times \vec{b}\] is a unit vector, then write the angle between \[\vec{a}\] and \[\vec{b}\] 

Answer 1

Let the angle between \[\vec{a}\] and \[\vec{b}\] be \[\theta\] It is given that \[\vec{a} \times \vec{b}\]  is a unit vector. 

\[\therefore \left| \vec{a} \times \vec{b} \right| = 1\]
\[ \Rightarrow \left| \vec{a} \right|\left| \vec{b} \right|\sin\theta = 1\]
\[ \Rightarrow 3 \times \frac{2}{3} \times \sin\theta = 1\]
\[ \Rightarrow \sin\theta = \frac{1}{2}\]
\[ \Rightarrow \theta = \frac{\pi}{6}\] 

\[\vec{a}\]  and \[\vec{b}\]  is \[\frac{\pi}{6}\]