Let $$\vec a = \hat i + 2\hat j + \hat k,\,\vec b = \hat i - \hat j + \hat k,\,\vec c = \hat i + \hat j - \hat k.$$         A vector in the plane of $$\vec a$$ and $$\vec b$$ whose projection on $$\vec c$$ is $$\frac{1}{{\sqrt 3 }},$$  is :

Solution :
A vector in the plane of $${\vec a}$$ and $${\vec b}$$ is
$$\vec u = \vec a + \lambda \vec b = \left( {1 + \lambda } \right)\hat i + \left( {2 - \lambda } \right)\hat j + \left( {1 + \lambda } \right)\hat k$$
Projection of $${\vec u}$$ on $$\vec c = \frac{1}{{\sqrt 3 }}\,\,\, \Rightarrow \frac{{\vec u.\vec c}}{{\left| {\vec c} \right|}} = \frac{1}{{\sqrt 3 }}$$
$$\eqalign{ & \Rightarrow \vec u.\vec c = 1\,\, \Rightarrow \left| {1 + \lambda + 2 - \lambda - 1 - \lambda } \right| = 1 \cr & \Rightarrow \left| {2 - \lambda } \right| = 1\,\, \Rightarrow \lambda = 1{\text{ or }}3 \cr & \Rightarrow \vec u = 2\hat i + \hat j + 2\hat k{\text{ or 4}}\hat i - \hat j + 4\hat k \cr} $$