Answer
$6\sqrt5$
Work Step by Step
We know that $ai+bj+ck=(a,b,c).$
We know from earlier that if two adjacent vectors forming the sides of a parallelogram are $u$ and $v$, then the area can be computed by $A=|u\times v|$.
We know that for a matrix
$
\left[\begin{array}{rrr}
a & b & c \\
d &e & f \\
g &h & i \\
\end{array} \right]
$
the determinant, $D=a(ei-fh)-b(di-fg)+c(dh-eg).$
$u×v$ is the determinant of the matrix $\begin{bmatrix}
i& j & k \\
3& 2&-1\\
1&2 &3 \\
\end{bmatrix}
$
Thus $u×v=(8,-10,4).$
Thus $A=|u\times v|=\sqrt{8^2+(-10)^2+4^2}=\sqrt{180}=6\sqrt5$
