Let $\overline{\mathrm{a}} = \overline{\mathrm{i}} + 2 \overline{\mathrm{j}} + \overline{\mathrm{k}}, \overline{\mathrm{b}} = \overline{\mathrm{i}} - \overline{\mathrm{j}}+\overline{\mathrm{k}} , \overline{\mathrm{c}}=\overline{\mathrm{i}}+\overline{\mathrm{j}}-\overline{\mathrm{k}} .$ A vector coplanar to $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ has a projection along $\overline{\mathrm{c}}$ of magnitude $\frac{1}{\sqrt{3}}$ then the

A vector $\overline{d}$ in the plane of $\overline{a}$ and $\overline{b}$ is given by

$d=\lambda a+\mu b=(\lambda +\mu )i+(2\lambda -\mu )j+(\lambda +\mu )k$

$Now the projection of d on c is$$\frac{(c\cdot d)}{\left|c\right|}=\frac{(\lambda +\mu +(2\lambda -\mu )-(\lambda +\mu \left)\right)}{\sqrt{3}}=\frac{1}{\sqrt{3}}$(given)

$2\lambda -\mu =1$. thus

$d=(3\lambda -1)i+j+(3\lambda -1)k$

$So for \lambda =-1,d=-4i+j-4k$

$So for \lambda =1,d=2i+j+2k$