Any vector that is coplanar with \(\vec{a}\) and \(\vec{c}\) is: \[ \vec{a} + \lambda \vec{c} = (2 + \lambda)\hat{i} + (1 + \lambda)\hat{j} + (2 - 2\lambda)\hat{k} \] The projection on \(\vec{b}\) is: \[ (\vec{a} + \lambda \vec{c}) \cdot \hat{b} = \frac{(2 + \lambda) - (1 + \lambda) + 2(2 - 2\lambda)}{\sqrt{1^2 + (-1)^2 + 2^2}} = \frac{5 - 4\lambda}{\sqrt{6}} \] Given that the projection on \(\vec{b}\) is of magnitude \(\frac{1}{\sqrt{6}}\): \[ 5 - 4\lambda = 1 \implies \lambda = 1 \] The vector is: \[ \vec{a} + \lambda \vec{c} = (2 + \lambda)\hat{i} + (1 + \lambda)\hat{j} + (2 - 2\lambda)\hat{k} = 3\hat{i} + 2\hat{j} \]