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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.
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Solution
\[\text{ Given that the angle between } \vec{a} \text{ and } \vec{b} {\text{ is } 30}^0 .\]
\[\text{ Also },\]
\[\left| \vec{a} \right| = \left| \vec{b} \right|; \vec{a} . \vec{b} = \frac{1}{2}\]
\[\text{ We know that }\]
\[ \vec{a} . \vec{b} = \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta\]
\[ \Rightarrow \frac{1}{2} = \left| \vec{a} \right|\left| \vec{a} \right| \cos 60\]
\[ \Rightarrow \frac{1}{2} = \left| \vec{a} \right|^2 \left( \frac{1}{2} \right)\]
\[ \Rightarrow \left| \vec{a} \right|^2 = 1\]
\[ \Rightarrow \left| \vec{a} \right| = 1\]
\[ \therefore \left| \vec{a} \right| = \left| \vec{b} \right| = 1\]
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