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प्रश्न
Let \[\vec{a} = 5 \hat{i} - \hat{j} + 7 \hat{k} \text{ and } \vec{b} = \hat{i} - \hat{j} + \lambda \hat{k} .\] Find λ such that \[\vec{a} + \vec{b}\] is orthogonal to \[\vec{a} - \vec{b}\]
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उत्तर
\[\text{ Given that }\]
\[ \vec{a} = 5 \hat{i} - \text{j} + 7 \hat{k} ; \vec{b} = \hat{i} - \hat{j} + \lambda \hat{k} \]
\[ \therefore \vec{a} + \vec{b} = 5 \hat{i} - \hat{j} + 7 \hat{k} + \hat{i} - \hat{j} + \lambda \hat{k} = 6 \hat{i} - 2 \hat{j} + \left( 7 + \lambda \right) \hat{k} \]
\[\text{ and } \vec{a} - \vec{b} = 5 \hat{i} - \hat{j} + 7 \hat{k} - \left( \hat{i} - \hat{j} + \lambda \hat{k} \right) = 4 \hat{i} + 0 \hat{j} + \left( 7 - \lambda \right) \hat{k} \]
\[\text{ Given that } \vec{a} + \vec{b} \text{ is orthogonal to } \vec{a} - \vec{b} . \]
\[ \Rightarrow \left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 0\]
\[ \Rightarrow \left[ 6\hat{i} - 2 \hat{j} + \left( 7 + \lambda \right) \hat{k} \right] . \left[ 4 \hat{i} + 0 \hat{j} + \left( 7 - \lambda \right) \hat{k} \right] = 0\]
\[ \Rightarrow 24 + 0 + 49 - \lambda^2 = 0\]
\[ \Rightarrow \lambda^2 = 73\]
\[ \Rightarrow \lambda = \sqrt{73}\]
