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Question
If \[\vec{p} = 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \text{ and } \vec{q} = \hat{i} + 3 \hat{j} - 5 \hat{k} ,\] then find the value of λ, so that \[\vec{p} + \vec{q}\] and \[\vec{p} - \vec{q}\] are perpendicular vectors.
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Solution
\[\text{Given that}\]
\[ \vec{p} = 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \]
\[\text{ and } \vec{q} = \hat{i} + 3 \hat{j} - 5 \hat{k} \]
\[ \vec{p} + \vec{q} = \left( 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \right) + \left( \hat{i} + 3 \hat{j} - 5 \hat{k} \right) = 6 \hat{i} + \left( \lambda + 3 \right) \hat{j} - 8 \hat{k} \]
\[ \vec{p} - \vec{q} = \left( 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \right) - \left( \hat{i} + 3 \hat{j} - 5 \hat{k} \right) = 4 \hat{i} + \left( \lambda - 3 \right) \hat{j} + 2 \hat{k} \]
\[\text{ Given that } \vec{p} + \vec{q} \text{ is orthogonal to } \vec{p} - \vec{q} . \]
\[ \Rightarrow \left( \vec{p} + \vec{q} \right) . \left( \vec{p} - \vec{q} \right) = 0\]
\[ \Rightarrow \left[ 6 {i} + \left( \lambda + 3 \right) \hat{j} - 8 \hat{k} \right] . \left[ 4 \hat{i} + \left( \lambda - 3 \right) \hat{j} + 2 \hat{k} \right] = 0\]
\[ \Rightarrow 24 + \lambda^2 - 9 - 16 = 0\]
\[ \Rightarrow \lambda^2 = 1\]
\[ \therefore \lambda = \pm 1\]
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