$\text{ Let }\overrightarrow{a}=\widehat{i}+2\widehat{j}+\widehat{k},\overrightarrow{b}=\widehat{i}-\widehat{j}+\widehat{k}\text{ and }\overrightarrow{c}=\widehat{i}+\widehat{j}-\widehat{k}$. A vector in the plane of $\overrightarrow{a}\text{ and }\overrightarrow{b}$whose projection on $\overrightarrow{c}\text{ is }1/\sqrt{3}$ is

$\text{ A vector in the plane of }\overrightarrow{a}\text{ and }\overrightarrow{b}\text{ is }$

$\overrightarrow{u}=\mu \overrightarrow{a}+\lambda \overrightarrow{b}=(\mu +\lambda )\widehat{i}+(2\mu -\lambda )\widehat{j}+(\mu +\lambda )\widehat{k}$

$\text{ Projection of }\overrightarrow{u}\text{ on }\overrightarrow{c}=\frac{1}{\sqrt{3}}$

$\Rightarrow \frac{\overrightarrow{u}\cdot \overrightarrow{c}}{\left|\overrightarrow{c}\right|}=\frac{1}{\sqrt{3}}$

$\begin{array}{l}\mathrm{or} \overrightarrow{\mathrm{u}}\cdot \overrightarrow{\mathrm{c}}=1\\ \mathrm{or} |\mathrm{\mu }+\mathrm{\lambda }+2\mathrm{\mu }-\mathrm{\lambda }-\mathrm{\mu }-\mathrm{\lambda }|=1\\ \mathrm{or} |2\mathrm{\mu }-\mathrm{\lambda }|=1\\ \mathrm{or} \mathrm{\lambda }=2\mathrm{\mu }\pm 1\\ \Rightarrow \overrightarrow{\mathrm{u}}=2\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\text{ or }4\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+4\widehat{\mathrm{k}}\end{array}$