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Question
Write the value of p for which \[\vec{a} = 3 \hat{i} + 2 \hat{j} + 9 \hat{k} \text{ and } \vec{b} = \hat{i} + p \hat{j} + 3 \hat{k}\] are parallel vectors .
Solution
We have
\[ \vec{a} = 3 \hat{i} + 2 \hat{j} + 9 \hat{k} \text{ and } \vec{b} = \hat{i} + p \hat{j} + 3 \hat{k} \]
\[\text{ Given that } \vec{a} \text{ and } \vec{b} \text{ are parallel }.\]
\[ \Rightarrow \vec{a} = t \vec{b} \text{ for some t. }\]
\[ \Rightarrow 3 \hat{i} + 2 \hat{j} + 9 \hat{k} = t \left( \hat{i}+ p \hat{j} + 3 \hat{k} \right)\]
\[ \Rightarrow 3 \hat{i} + 2 \hat{j} + 9 \hat{k} = t \hat{i} + pt \hat{j} + 3t \hat{k} \]
\[\text{ Comparing both sides, we get }\]
\[3 = t, 2 = pt \text{ and } 9 = 3t\]
\[ \Rightarrow t = 3 \text{ and }pt = 2\]
\[ \Rightarrow 3t = 2\]
\[ \therefore t = \frac{2}{3}\]
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