If $\overrightarrow{a}=\hat i+\hat j+\hat k$ and $\overrightarrow{b}=\hat j-\hat k$,find a vector $\overrightarrow{c}$ such that $\overrightarrow{a}\times\overrightarrow{c}=\overrightarrow{b}$ and $\overrightarrow{a}.\overrightarrow{c}=3$.

Question

$\begin{array}{1 1} (A)\;\large\frac{3}{5} \hat i+\large\frac{3}{2} \hat j+\large\frac{3}{2} \hat k \\(B)\;\large\frac{5}{8} \hat i+\large\frac{2}{8} \hat j+\large\frac{2}{7} \hat k \\ (C)\;\large\frac{5}{3} \hat i+\large\frac{2}{3} \hat j+\large\frac{2}{3} \hat k \\ (D)\;\large\frac{1}{3} \hat i+\large\frac{1}{3} \hat j+\large\frac{1}{3} \hat k \end{array}$