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Question
Solution
\[\text{ Given } : \]
\[ \vec{a} = 3 \hat{ i } - \hat{ j } - 2 \hat{ k } \]
\[ \vec{b} = 2 \hat{ i } + 3 \hat{ j } + \hat{ k } \]
\[ \therefore \vec{a} + 2 \vec{b} = 3 \hat{ i } - \hat{ j } - 2 \hat{ k } + 2 \left( 2 \hat{ i } + 3 \hat{ j } + \hat{ k } \right)\]
\[ = 7 \hat{ i } + 5 \hat{ j } + 0 \hat{ k } \]
\[ \therefore 2 \vec{a} - \vec{b} = 2 \left( 3 \hat{ i } - \hat{ j } - 2 \hat{ k } \right) - \left( 2 \hat{ i } + 3 \hat{ j } + \hat{ k } \right)\]
\[ = 4 \hat{ i } - 5 \hat{ j } - 5 \hat{ k } \]
\[\left( \vec{a} + 2 \vec{b} \right) \times \left( 2 \vec{a} - \vec{b} \right) = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 7 & 5 & 0 \\ 4 & - 5 & - 5\end{vmatrix}\]
\[ = \hat{ i } \left( - 25 + 0 \right) - \hat{ j } \left( - 35 + 0 \right) + \hat{ k } \left( - 35 - 20 \right)\]
\[ = - 25 \hat{ i }+ 35 \hat { j } - 55 \hat{ k } \]
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