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# If → a = 2 ^ I − ^ J + ^ K → B = ^ I + ^ J − 2 ^ K → C = ^ I + 3 ^ J − ^ K Find λ Such that → a is Perpendicular to λ → B + → C - CBSE (Science) Class 12 - Mathematics

ConceptBasic Concepts of Vector Algebra

#### Question

If $\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}$  $\vec{b} = \hat{i} + \hat{j} - 2 \hat{k}$  $\vec{c} = \hat{i} + 3 \hat{j} - \hat{k}$ find λ such that $\vec{a}$ is perpendicular to $\lambda \vec{b} + \vec{c}$

#### Solution

The given vectors are $\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}$  $\vec{b} = \hat{i} + \hat{j} - 2 \hat{k}$  and  $\vec{c} = \hat{i} + 3 \hat{j} - \hat{k}$

Now,

$\lambda \vec{b} + \vec{c} = \lambda\left( \hat{i} + \hat{j} - 2 \hat{k} \right) + \left( \hat{i} + 3 \hat{j} - \hat{k} \right) = \left( \lambda + 1 \right) \hat{i} + \left( \lambda + 3 \right) \hat{j} - \left( 2\lambda + 1 \right) \hat{k}$ It is given that

$\vec{a} \perp \left( \lambda \vec{b} + \vec{c} \right)$

$\Rightarrow \vec{a} . \left( \lambda \vec{b} + \vec{c} \right) = 0$

$\Rightarrow \left( 2 \hat{i} - \hat{j} + \hat{k} \right) . \left[ \left( \lambda + 1 \right) \hat{i} + \left( \lambda + 3 \right) \hat{j} - \left( 2\lambda + 1 \right) \hat{k} \right] = 0$

$\Rightarrow 2\left( \lambda + 1 \right) - \left( \lambda + 3 \right) - \left( 2\lambda + 1 \right) = 0$

$\Rightarrow 2\lambda + 2 - \lambda - 3 - 2\lambda - 1 = 0$

$\Rightarrow \lambda = - 2$

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Solution If → a = 2 ^ I − ^ J + ^ K → B = ^ I + ^ J − 2 ^ K → C = ^ I + 3 ^ J − ^ K Find λ Such that → a is Perpendicular to λ → B + → C Concept: Basic Concepts of Vector Algebra.
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