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. [CBSE 2014]

Answer 1

Let $\overrightarrow{a}=3\hat{i}+2\hat{j}+9\hat{k}$ and $\overrightarrow{b}=\hat{i}-2p\hat{j}+3\hat{k}$ be the two given vectors.

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are parallel, then

$\overrightarrow{b}=\lambda \overrightarrow{a}$ for some scalar λ

$$\begin{gathered} \therefore \hat{i}-2p\hat{j}+3\hat{k}=\lambda \left(3\hat{i}+2\hat{j}+9\hat{k}\right) \\ \Rightarrow \hat{i}-2p\hat{j}+3\hat{k}=3\lambda \hat{i}+2\lambda \hat{j}+9\lambda \hat{k} \\ \Rightarrow 1=3\lambda \mathrm{and}-2p=2\lambda \left(\mathrm{Equating}\mathrm{coefficients}\mathrm{of}\hat{i},\hat{j},\hat{k}\right) \\ \Rightarrow p=-\lambda =-\frac{1}{3} \end{gathered}$$

Thus, the value of p is $-\frac{1}{3}$.