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# If → a = 2 ^ I + 2 ^ J + 3 ^ K , → B = − ^ I + 2 ^ J + ^ K and → C = 3 ^ I + ^ J → a + λ → B is Perpendicular to → C Then Find the Value of λ. - CBSE (Arts) Class 12 - Mathematics

ConceptBasic Concepts of Vector Algebra

#### Question

If $\vec{a} = 2 \hat{i} + 2 \hat{j} + 3 \hat{k} , \vec{b} = - \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{c} = 3 \hat{i} + \hat{j}$ $\vec{a} + \lambda \vec{b}$ is perpendicular to $\vec{c}$ then find the value of λ.

#### Solution

$\text{ We have }$
$\vec{a} = 2 \hat{i} + 2 \hat{j} + 3 \hat{k}$
$\vec{b} = - \hat{i} +2 \hat{j} + \hat{k}$
$\text{and}$
$\vec{c} = 3 \hat{i} + \hat{j}$
$\therefore \vec{a} + \lambda \vec{b} = 2 \hat{i} + 2 \hat{j} + 3 \hat{k} + \lambda \left( - \hat{i} + 2 \hat{j} + \hat{k} \right) = \left( 2 - \lambda \right) \hat{i} + \left( 2 + 2\lambda \right) \hat{j} + \left( 3 + \lambda \right) \hat{k}$
$\text{ Given that } \vec{a} + \lambda \vec{b} \text{ is perpendicular to } \vec{c} .$
$\Rightarrow \left( \vec{a} + \lambda \vec{b} \right) . \vec{c} = 0$
$\Rightarrow \left[ \left( 2 - \lambda \right)\hat{i} + \left( 2 + 2\lambda \right) \hat{j} + \left( 3 + \lambda \right) \hat{k} \right] . \left( 3 \hat{i} + \hat{j} + 0 \hat{k} \right) = 0$
$\Rightarrow 3 \left( 2 - \lambda \right) + 1 \left( 2 + 2\lambda \right) + 0 = 0$
$\Rightarrow 6 - 3\lambda + 2 + 2\lambda = 0$
$\therefore \lambda = 8$

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