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Dot Products of a Vector with Vectors ^ I − ^ J + ^ K , 2 ^ I + ^ J − 3 ^ K and I + ^ J + ^ K Are Respectively 4, 0 and 2. Find the Vector. - CBSE (Arts) Class 12 - Mathematics

ConceptBasic Concepts of Vector Algebra

Question

Dot products of a vector with vectors $\hat{i} - \hat{j} + \hat{k} , 2\hat{ i} + \hat{j} - 3\hat{k} \text{ and } \text{i} + \hat{j} + \hat{k}$  are respectively 4, 0 and 2. Find the vector.

Solution

$\text{ Let }a \hat{i} + b \hat{j} + c\hat{k} \text{ be the required vector }.$
$\text{ Given that }$
$\left( a\hat{i} + b\hat{j} + c\hat{k} \right) . \left(\hat{i} - \hat{j} + \hat{k} \right) = 4$
$\Rightarrow a - b + c = 4 . . . \left( 1 \right)$
$\left( a \hat{i} + b \hat{j}+ c{k} \right) . \left( 2 \hat{i} + \hat{j} - 3\hat{k}\right) = 0$
$\Rightarrow 2a + b - 3c = 0 . . . \left( 2 \right)$
$\left( a \hat{i} + b\hat{j} + c\hat{k}\right) . \left( \hat{i} + \hat{j} + \hat{k} \right) = 2$
$\Rightarrow a + b + c = 2 . . . \left( 3 \right)$
$\text{ Solving } (1), (2) \text{ and } (3), \text{we get}$
$a = 2; b = - 1; c = 1$
$\text{So},a \hat{i} + b \hat{j} + c\hat{k} = 2 \hat{i} - \hat{j} + \hat{k}$

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Solution Dot Products of a Vector with Vectors ^ I − ^ J + ^ K , 2 ^ I + ^ J − 3 ^ K and I + ^ J + ^ K Are Respectively 4, 0 and 2. Find the Vector. Concept: Basic Concepts of Vector Algebra.
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