#### Question

Find the magnitude of two vectors \[\vec{a} \text{ and } \vec{b}\] that are of the same magnitude, are inclined at 60° and whose scalar product is 1/2.

#### Solution

\[\text{ Given that the angle between } \vec{a} \text{ and } \vec{b} {\text{ is } 30}^0 .\]

\[\text{ Also },\]

\[\left| \vec{a} \right| = \left| \vec{b} \right|; \vec{a} . \vec{b} = \frac{1}{2}\]

\[\text{ We know that }\]

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

\[ \Rightarrow \frac{1}{2} = \left| \vec{a} \right|\left| \vec{a} \right| \cos 60\]

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

\[ \Rightarrow \left| \vec{a} \right|^2 = 1\]

\[ \Rightarrow \left| \vec{a} \right| = 1\]

\[ \therefore \left| \vec{a} \right| = \left| \vec{b} \right| = 1\]