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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.
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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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