Scaling factor for the area of the parallelogram formed by two vectors
# 2D
$
\det\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)=\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc
$
# 3D
Sign convention given as:
$
\begin{vmatrix}+&-&+\\-&+&-\\+&-&+\end{vmatrix}
$
$
\begin{align}\det\left(\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\right)&=\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}\\&=a(ei-fh)-b(di-fg)+c(dh-eg)
\end{align}
$