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# 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 (Arts) Class 12 - Mathematics

ConceptBasic Concepts of Vector Algebra

#### 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

$\text{Given that}$

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

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

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

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

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

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

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

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

$\Rightarrow \lambda^2 = 1$

$\therefore \lambda = \pm 1$

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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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