Let a= i +2 j + k , b= i - j + k , c= i + j - k . A vector coplanar...

Let vector $\mathbf{r}$ be coplanar to $\mathbf{a} and$\mathbf{b}$.$$\begin{array}{ll} \therefore& \mathbf{r}=\mathbf{a}+ t \mathbf{b} \\ \Rightarrow & \mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})+t(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=\hat{\mathbf{i}}(1+t)+\hat{\mathbf{j}}(2-t)+\hat{\mathbf{k}}(1+t) \end{array}$$The projection of$\mathbf{r}$on$\mathbf{c}=\frac{1}{\sqrt{3}} \Rightarrow \frac{\mathbf{r} \cdot \mathbf{c}}{|\mathbf{c}|}=\frac{1}{\sqrt{3}}\Rightarrow \frac{|1 \cdot(1+t)+1 \cdot(2-t)-1 \cdot(1+t)|}{\sqrt{3}}=\frac{1}{\sqrt{3}}\Rightarrow|2-t|=\pm 1 \Rightarrow t=1$or 3 When,$t=1$we have$\mathbf{r}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$When,$t=3$we have$\mathbf{r}=4 \hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}}$