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Question
Find a unit vector perpendicular to each of the vectors \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} , \text{ where } \vec{a} = 3 \hat{ i } + 2 \hat{ j } + 2 \hat{ k } \text{ and } \vec{b} = \hat{ i } + 2 \hat{ j } - 2 \hat{ k } .\]
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Solution
\[\text{ Given } : \]
\[ \vec{a} = 3 \hat{ i } + 2 \hat{ j } + 2 \hat{ k } \]
\[ \vec{b} = \hat{ i } + 2 \hat{ j } - 2 \hat{ k } \]
\[ \therefore \vec{a} + \vec{b} = 4 \hat{ i } + 4 \hat{ j } + 0 \hat{ k } \]
\[ \vec{a} - \vec{b} = 2 \hat{ i } + 0 \hat{ j } + 4 \hat{ k } \]
\[\left( \vec{a} + \vec{b} \right) \times \left( \vec{a} - \vec{b} \right) = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 4 & 4 & 0 \\ 2 & 0 & 4\end{vmatrix}\]
\[ = 16 \hat{ i } - 16 \hat{ j } - 8 \hat{ k } \]
\[ \therefore \left| \left( \vec{a} + \vec{b} \right) \times \left( \vec{a} - \vec{b} \right) \right| = \sqrt{256 + 256 + 64}\]
\[ = \sqrt{576}\]
\[ = 24\]
\[\text{ Unit vector perpendicular to both } \vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} =\frac{\left( \vec{a} + \vec{b} \right) \times \left( \vec{a} - \vec{b} \right)}{\left| \left( \vec{a} + \vec{b} \right) \times \left( \vec{a} - \vec{b} \right) \right|}\]
\[ = \frac{16 \hat{ i } - 16 \hat{ j } - 8 \hat{ k } }{24}\]
\[ = \frac{8 \left( 2 \hat{ i } - 2 \hat{ j } - \hat{ k } \right)}{24}\]
\[ = \frac{1}{3}\left( 2 \hat{ i } - 2 \hat{ j } - \hat{ k } \right)\]
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