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Question
If \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} . \vec{b} = 2, \text{ find } \left| \vec{a} - \vec{b} \right| .\]
Solution
\[\text{ We have }\]
\[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} . \vec{b} = 2\]
\[\text{ Now }, \]
\[ \left| \vec{a} - \vec{b} \right|^2 = \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 - 2 \vec{a} . \vec{b} \]
\[ = 2^2 + 5^2 - 2 \left( 2 \right)\]
\[ = 4 + 25 - 4\]
\[ = 25\]
\[ \therefore \left| \vec{a} - \vec{b} \right| = \sqrt{25} = 5\]
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