Question

If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} . \vec{b} \right| = \left| \vec{a} \times \vec{b} \right|,\]  write the angle between \[\vec{a} \text{ and } \vec{b} .\]

 

 

 

Answer 1

\[\text{ Let } \theta \text{ be the angle between
} a^\to \text{ and }  \vec{b} . \]

\[\text{ We know } \]

\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \left| \sin \theta \right|\]

\[\left| \vec{a} . \vec{b} \right| = \left| \vec{a} \right| \left| \vec{b} \right|\left| \cos \theta \right|\]

\[\text{ Now, } \]

\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{a} . \vec{b} \right| (\text{Given } )\]

\[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right| \left| \sin \theta \right| = \left| \vec{a} \right| \left| \vec{b} \right| \left| \cos \theta \right|\]

\[ \Rightarrow \left| \sin \theta \right| = \left| \cos \theta \right|\]

\[ \Rightarrow \theta = \frac{\pi}{4}\]