If \(\vec{a}=\hat{i}+\hat{j}+\hat{k}\), \(\vec{b}=2\hat{i}-\hat{j}+3\hat{k}\) and \(\vec{c}=\hat{i}-2\hat{j}+\hat{k}\), and a unit vector parallel to the vector \(2\vec{a}-\vec{b}+3\vec{c}\) ?
1
\(\frac{9}{{\sqrt {22} }}\hat{i}-\frac{3}{{\sqrt {22} }}\hat{j}+\frac{1}{{\sqrt {22} }}\hat{k}\)
2
\(\frac{2}{{\sqrt {22} }}\hat{i}-\frac{3}{{\sqrt {22} }}\hat{j}+\frac{2}{{\sqrt {22} }}\hat{k}\)
3
\(\frac{3}{{\sqrt {22} }}\hat{i}-\frac{3}{{\sqrt {22} }}\hat{j}+\frac{3}{{\sqrt {22} }}\hat{k}\)
4
\(\frac{3}{{\sqrt {22} }}\hat{i}-\frac{3}{{\sqrt {22} }}\hat{j}+\frac{2}{{\sqrt {22} }}\hat{k}\)