If 4i+2p/3 j+pk is parallel to the vector i+2j+3k, find p.

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \overrightarrow{\mathrm{a}}=4\hat{\mathrm{i}}+\frac{2\mathrm{p}}{3}\hat{\mathrm{j}}+\mathrm{p}\hat{\mathrm{k}} \\ \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}} \\ \mathrm{As} \mathrm{given} \mathrm{vectors} \mathrm{are} \mathrm{parallel}, \mathrm{hence} \mathrm{they} \mathrm{can} \mathrm{be} \mathrm{written} \mathrm{as}: \\ \overrightarrow{\mathrm{a}}=\mathrm{\lambda }\overrightarrow{\mathrm{b}}, \mathrm{where} \mathrm{\lambda } \mathrm{is} \mathrm{constant}. \\ \Rightarrow 4\hat{\mathrm{i}}+\frac{2\mathrm{p}}{3}\hat{\mathrm{j}}+\mathrm{p}\hat{\mathrm{k}}=\mathrm{\lambda }\left(\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}}\right) \\ \Rightarrow 4\hat{\mathrm{i}}+\frac{2\mathrm{p}}{3}\hat{\mathrm{j}}+\mathrm{p}\hat{\mathrm{k}}=\mathrm{\lambda }\hat{\mathrm{i}}+2\mathrm{\lambda }\hat{\mathrm{j}}+3\mathrm{\lambda }\hat{\mathrm{k}} \\ \mathrm{On} \mathrm{comparing} \mathrm{respective} \mathrm{coefficients}, \mathrm{we} \mathrm{get}: \\ 4=\mathrm{\lambda } \\ \frac{2\mathrm{p}}{3}=2\mathrm{\lambda } \\ \Rightarrow \mathrm{p}=3\mathrm{\lambda } \\ \mathrm{Put} \mathrm{\lambda }=4 \\ \Rightarrow \mathrm{p}=12 \\ \mathrm{which} \mathrm{should} \mathrm{satisify} \mathrm{equation} \mathrm{obtained} \mathrm{by} \mathrm{comapring} \mathrm{coefficient} \mathrm{of} \hat{\mathrm{k}}. \\ \mathrm{Comparing} \mathrm{coefficient} \mathrm{of} \hat{\mathrm{k}}, \mathrm{we} \mathrm{get} \\ \mathrm{p}=3\mathrm{\lambda }=3\times 4=12 \mathrm{which} \mathrm{is} \mathrm{already} \mathrm{obtained}. \\ \mathrm{Hence} \\ \boxed{\mathbf{ }\mathbf{p}\mathbf{=}\mathbf{12}\mathbf{ }} \end{gathered}$$

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