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Question
If \[\vec{a}\] and \[\vec{b}\] are perpendicular vectors, \[\left| \vec{a} + \vec{b} \right| = 13\] and \[\left| \vec{a} \right| = 5\] find the value of \[\left| \vec{b} \right|\]
Solution
\[\left| \vec{a} + \vec{b} \right| = 13\] has been taken in order to solve the question. It is given that \[\vec{a}\] and \[\vec{b}\] are perpendicular vectors.
\[\therefore \vec{a} . \vec{b} = 0\]
\[\left| \vec{a} + \vec{b} \right| = 13\]
\[ \Rightarrow \left| \vec{a} + \vec{b} \right|^2 = 169\]
\[ \Rightarrow \left| \vec{a} \right|^2 + 2 \vec{a} . \vec{b} + \left| \vec{b} \right|^2 = 169\]
\[ \Rightarrow 25 + 2 \times 0 + \left| \vec{b} \right|^2 = 169............. \left[ \text{ Using }\left( 1 \right) \right]\]
\[\Rightarrow \left| \vec{b} \right|^2 = 169 - 25 = 144\]
\[ \Rightarrow \left| \vec{b} \right| = 12\]
Thus, the value of \[\left| \vec{b} \right|\] is 12
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