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Question
\[\text{ If } \vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + 2\hat{k} , \text{find} \left( \vec{a} - 2 \vec{b} \right) \cdot \left( \vec{a} + \vec{b} \right) .\]
Solution
\[\text{We have}\]
\[ \vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + 2 \hat{k} \]
\[ \vec{a} - 2 \vec{b} = \left( \hat{i} - \hat{j}\right) - 2 \left( - \hat{j} + 2 \hat{k} \right) = \hat{i} - \hat{j} + 2 \hat{j} - 4 \hat{k} = \hat{i} + \hat{j} - 4 \hat{k} \]
\[ \vec{a} + \vec{b} = \hat{i} -\hat{j} - \hat{j} + 2 \hat{k}^\ = \hat{i}- 2 \hat{j} + 2 \hat{k} \]
\[\left( \vec{a} - 2 \vec{b} \right) . \left( \vec{a} + \vec{b} \right)\]
\[ = \left( \hat{i} + \hat{j} - 4 \hat{k} \right) . \left( \hat{i}^ - 2 \hat{j} + 2 \hat{k} \right)\]
\[ = 1 - 2 - 8\]
\[ = - 9\]
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