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Solution for If → P = 5 ^ I + λ ^ J − 3 ^ K and → Q = ^ I + 3 ^ J − 5 ^ K , Then Find the Value of λ, So that → P + → Q and → P − → Q Are Perpendicular Vectors. - CBSE (Commerce) Class 12 - Mathematics

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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. 

Solution

\[\text{Given that}\]

\[ \vec{p} = 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \]

\[\text{ and } \vec{q} = \hat{i} + 3 \hat{j} - 5 \hat{k} \]

\[ \vec{p} + \vec{q} = \left( 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \right) + \left( \hat{i} + 3 \hat{j} - 5 \hat{k} \right) = 6 \hat{i} + \left( \lambda + 3 \right) \hat{j} - 8 \hat{k} \]

\[ \vec{p} - \vec{q} = \left( 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \right) - \left( \hat{i} + 3 \hat{j} - 5 \hat{k} \right) = 4 \hat{i} + \left( \lambda - 3 \right) \hat{j} + 2 \hat{k} \]

\[\text{ Given that } \vec{p} + \vec{q} \text{ is orthogonal to } \vec{p} - \vec{q} . \]

\[ \Rightarrow \left( \vec{p} + \vec{q} \right) . \left( \vec{p} - \vec{q} \right) = 0\]

\[ \Rightarrow \left[ 6 hat{i} + \left( \lambda + 3 \right) \hat{j} - 8 \hat{k} \right] . \left[ 4 \hat{i} + \left( \lambda - 3 \right) \hat{j} + 2 \hat{k} \right] = 0\] 

\[ \Rightarrow 24 + \lambda^2 - 9 - 16 = 0\]

\[ \Rightarrow \lambda^2 = 1\]

\[ \therefore \lambda = \pm 1\] 

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Solution If → P = 5 ^ I + λ ^ J − 3 ^ K and → Q = ^ I + 3 ^ J − 5 ^ K , Then Find the Value of λ, So that → P + → Q and → P − → Q Are Perpendicular Vectors. Concept: Basic Concepts of Vector Algebra.
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