If the projection of the vector $\displaystyle \hat{i}+\hat{j}+\hat{k}$ on the vector $\displaystyle \mathrm{p} \hat{i}+\hat{j}-2 \hat{k}$ is $\displaystyle \frac{1}{3}$, then find the value(s) of p.

Here, $\displaystyle \left[\frac{(\hat{\imath}+\hat{\jmath}+\hat{k}) \cdot(\mathrm{p} \hat{\imath}+\hat{\jmath}-2 \hat{k})}{\sqrt{\mathrm{p}^{2}+1+4}}\right]=\frac{1}{3}$\Rightarrow \frac{\mathrm{p}-1}{\sqrt{\mathrm{p}^{2}+5}}=\frac{1}{3}$\displaystyle \Rightarrow 8 \mathrm{p}^{2}-18 \mathrm{p}+4=0$ $\displaystyle 4 \mathrm{p}^{2}-9 \mathrm{p}+2=0$ $\displaystyle 4 \mathrm{p}^{2}-8 \mathrm{p}-\mathrm{p}+2=0$ $\displaystyle (4 \mathrm{p}-1)(\mathrm{p}-2)=0$ $\displaystyle \Rightarrow \mathrm{p}=2$ or $\displaystyle \mathrm{p}=\frac{1}{4}$