Question

If \[\vec{a} \text{ and } \vec{b}\] are two vectors of the same magnitude inclined at an angle of 30°, such that \[\vec{a} \cdot \vec{b} = 3, \text{ find } \left| \vec{a} \right|, \left| \vec{b} \right| .\] 

Answer 1

\[\text{ Given that the angle between } \vec{a} \text{ and } \vec{ b }\text { is 30 }^0 .\]
\[\text{ Also },\]
\[\left| \vec{a} \right| = \left| \vec{b} \right| \text{ and } \vec{a} . \vec{b} = 3\]
\[\text{ We know that }\]
\[ \vec{a} . \vec{b} = \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta\]
\[ \Rightarrow 3 = \left| \vec{a} \right|\left| \vec{a} \right| \cos 30\]
\[ \Rightarrow 3 = \left| \vec{a} \right|^2 \left( \frac{\sqrt{3}}{2} \right)\]
\[ \Rightarrow \left| \vec{a} \right|^2 = \frac{6}{\sqrt{3}} = 2\sqrt{3}\]
\[ \Rightarrow \left| \vec{a} \right| = \sqrt{2\sqrt{3}} = \left| \vec{b} \right|\]
\[ \therefore \left| \vec{a} \right| = \left| \vec{b} \right| = \sqrt{2\sqrt{3}}\]
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