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Question
Dot product of a vector with \[\hat{i} + \hat{j} - 3\hat{k} , \hat{i} + 3\hat{j} - 2 \hat{k} \text{ and } 2 \hat{i} + \hat{j} + 4 \hat{k}\] are 0, 5 and 8 respectively. Find the vector.
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Solution
\[\text{Let a } \hat{i} + b\hat{j}+ c \hat{k}\text{ be the required vector. }\]
\[\text{Given that}\]
\[\left( a\hat{i} + b \hat{j} + c \hat{k}\right) . \left( \hat{i} + \hat{j} - 3 \hat{k} \right) = 0\]
\[ \Rightarrow a + b - 3c = 0 . . . \left( 1 \right)\]
\[\left( \hat{ai} + \hat{bj} + \hat{ck}\right) . \left(\hat{i} + 3 \hat{j} - \hat{2k}\right) = 5\]
\[ \Rightarrow a + 3b - 2c = 5 . . . \left( 2 \right)\]
\[\left( \hat{ai} + b \hat{j} + \hat{ck} \right) . \left( \hat{2i} + \hat{j} + \hat{4k} \right) = 5\]
\[ \Rightarrow 2a + b + 4c = 8 . . . \left( 3 \right)\]
\[\text{ Solving } (1), (2) \text{ and } (3), \text{ we get }\]
\[a = 1, b = 2, c = 1\]
\[\text{ So },a \hat{i} + \hat{bj} + \hat{ck} = \hat{i} + \hat{2j} + \hat{k} \]
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