Official solution
\[\& 3=\frac{|[(\lambda -1)\hat{i}+3\hat{j}+3\hat{k}].(\hat{k}\times \hat{j})|}{|\hat{k}\times \hat{j}|}=|\lambda -1|=3 \ \& \Rightarrow \lambda =4-2,{\lambda }_{2}\]
\[\begin{gathered} (3,-4,4-k)\perp \hat{k} \\ 4-k=0\Rightarrow k=4 \\ \Rightarrow (4,-2,7)(1,2,7) \\ \Rightarrow \sqrt{{3}^{2}+{4}^{2}}=5 \end{gathered}\]