\(\vec a\) = 3\(\hat i\) - \(\hat j\) + 2\(\hat k\),
\(\vec b\) = 2\(\hat i\) + \(\hat j\) - 2\(\hat k\),
\(\vec a\)x \(\vec b\) = \(\begin{vmatrix}
\hat i & \hat j & \hat k \\[0.3em]
3 & -1 &2 \\[0.3em]
2 & 1 & -2
\end{vmatrix}\)
= \(\hat i\)\(\begin{vmatrix}
-1 &2 \\[0.3em]
1 & -2
\end{vmatrix}\) - \(\hat j\) \(\begin{vmatrix}
-3 &2 \\[0.3em]
1 & -2
\end{vmatrix}\) + \(\hat k\) \(\begin{vmatrix}
-3 &-1 \\[0.3em]
2 &1
\end{vmatrix}\)
= \(\hat i\) (2 - 2) - \(\hat j\) (- 6 - 4) + \(\hat k\) (3 + 2)
= 10\(\hat j\) + 5\(\hat k\)
