If \( \vec{A}=\hat{i}-\hat{j}+\hat{k} \) and \( \vec{B}=\sqrt{3} \hat{i}+3 \hat{j}-2 \hat{k} \) are the adjacent sides of a triangle, what is the area of the triangle in square unit?

A \( \frac{\sqrt{20+10 \sqrt{3}}}{2} \)

B \( \frac{1}{2} \left ( \sqrt{22 + \sqrt{3}} \right ) \)

C \( \frac{1}{2} \left ( \sqrt{22 - \sqrt{3}} \right ) \)

D \( \frac{1}{\sqrt{2}} \left ( \sqrt{22 + \left ( \sqrt{3} \right )} \right ) \)

Correct Answer: Option A