If the vectors \(\vec{a}=\lambda \hat{i}+\mu \hat{j}+4\hat{k},\vec{b}=−2\hat{i}+4\hat{j}−2\hat{k}\text{ }\) and \(\vec{c…

If the vectors \(\vec{a}=\lambda \hat{i}+\mu \hat{j}+4\hat{k},\vec{b}=−2\hat{i}+4\hat{j}−2\hat{k}\text{ }\) and \(\vec{c}=2\hat{i}+3\hat{j}+\hat{k}\) are coplanar and the projection of \(\vec{a}\) on the vector \(\vec{b}\) is \(\sqrt{54}\) then the sum of all possible values of \(\lambda +\mu\) is equal to

[JEE Main 2023, 29 Jan (Shift 1)]