Question

Write the value of p for which $\overrightarrow{a}=3\hat{i}+2\hat{j}+9\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}+p\hat{j}+3\hat{k}$ are parallel vectors.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have} \\ \overrightarrow{a}=3\hat{i}+2\hat{j}+9\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}+p\hat{j}+3\hat{k} \\ \text{Given that}\overrightarrow{a}\text{and}\overrightarrow{b}\text{are parallel.} \\ \Rightarrow \overrightarrow{a}=t\overrightarrow{b}\text{for some}\mathit{\text{t}}\text{.} \\ \Rightarrow 3\hat{i}+2\hat{j}+9\hat{k}=t\left(\hat{i}+p\hat{j}+3\hat{k}\right) \\ \Rightarrow 3\hat{i}+2\hat{j}+9\hat{k}=t\hat{i}+pt\hat{j}+3t\hat{k} \\ \mathrm{Comparing}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ 3=t,2=pt\text{and}9=3t \\ \Rightarrow t=3\text{and}pt=2 \\ \Rightarrow 3t=2 \\ \therefore t=\frac{2}{3} \end{gathered}$$