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Question
If the vectors \[3 \hat{i} - 2 \hat{j} - 4 \hat{k}\text{ and } 18 \hat{i} - 12 \hat{j} - m \hat{k}\] are parallel, find the value of m.
Solution
\[\text{ The given vectors are parallel }.\]
\[ \therefore 3 \hat{i} - 2 \hat{j} - 4 \hat{k} = t \left( 18 \hat{i} - 12 \hat{j} - m \hat{k} \right)\]
\[ \Rightarrow 3 \hat{i} - 2 \hat{j} - 4 \hat{k} = 18t \hat{i} - 12t \hat{j} - tm \hat{k} \]
\[\text{ Comparing both sides, we get }\]
\[ 18t = 3, - 12t = - 2, - 4 = - tm\]
\[ \Rightarrow t = \frac{1}{6} \]
\[\text{ Substituting the value of m in } -4=-tm, \text{ we get }\]
\[ - 4 = - m\left( \frac{1}{6} \right)\]
\[ \therefore m = 24\]
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