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Question
If \[\vec{a} = 5 \hat{i} - \hat{j} - 3 \hat{k} \text{ and } \vec{b} = \hat{i} + 3 \hat{j} - 5 \hat{k} ,\] then show that the vectors \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} \] are orthogonal.
Solution
\[\text{ Given that }\]
\[ \vec{a} = 5 \hat{i} - \hat{j} - 3 \hat{k} ; \vec{b} = \hat{i} + 3 \hat{j} - 5 \hat{k} \]
\[ \therefore \vec{a} + \vec{b} = 5 \hat{i} - \hat{j} - 3 \hat{k} + \hat{i} + 3 \hat{j} - 5 \hat{k} = 6 \hat{i} + 2 \hat{j} - 8 \hat{k} \]
\[\text{ And } \vec{a} - \vec{b} = 5 \hat{i} - \hat{j} - 3 \hat{k} - \left( \hat{i} + 3 \hat{j} - 5 \hat{k} \right) = 4 \hat{i} - 4 \hat{j} + 2 \hat{k} \]
\[\text{ Now },\]
\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right)\]
\[ = \left( 6 \hat{i}] + 2 \hat{j} - 8 \hat{k} \right) . \left( 4 \hat{i} - 4 \hat{j} + 2 \hat{k} \right)\]
\[ = 24 - 8 - 16\]
\[ = 0\]
\[So, \vec{a} + \vec{b} \text{ is orthogonal to } \vec{a} - \vec{b} .\]
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