Let \(\vec{a}=\hat{j}-\hat{k}\) and\(\vec{c}=\hat{i}-\hat{j}-\hat{k} \) . Then the vector \(\vec{b}\) satisfying \(\vec{a} \times \vec{b}+\vec{c}=\overrightarrow{0} \ and \ \vec{a} . \vec{b}=3\) is :

(1)\(-\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}\)

(2) \(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}\)

(3)\(\hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}}\)

(4) \(\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}\)