Question

Find the value of 'p' for which the vectors \[3 \hat{i} + 2 \hat{j} + 9 \hat{k}\] and \[\hat{i} - 2p \hat{j} + 3 \hat{k}\] are parallel.

Answer 1

Let \[3 \hat{i} + 2 \hat{j} + 9 \hat{k}\] and \[\hat{i} - 2p \hat{j} + 3 \hat{k}\] be the two given vectors.
If \[\overrightarrow{a}\] and \[\overrightarrow{b}\] are parallel, then \[\overrightarrow{b} = \lambda \overrightarrow{a}\] for some scalar λ 

\[\therefore \hat{i} - 2p \hat{j} + 3 \hat{k} = \lambda\left( 3 \hat{i} + 2 \hat{j} + 9 \hat{k} \right)\]

\[ \Rightarrow \hat{i} - 2p \hat{j} + 3 \hat{k} = 3\lambda \hat{i} + 2\lambda \hat{j} + 9\lambda \hat{k} \]

\[ \Rightarrow 1 = 3\lambda\text{ and }- 2p = 2\lambda \left( \text{ Equating coefficients of }\hat{i} , \hat{j} , \hat{k} \right)\]

\[ \Rightarrow p = - \lambda = - \frac{1}{3}\]
Thus, the value of p is \[- \frac{1}{3}\].