Example 2

Show that the area of the parallelogram constructed by vectors $AB$ and $AC$,with $A( a_1, a_2), B(b_1, b_2)$ and $C( c_1, c_2)$ is

$$\det(A)= \det( \left[ \begin{array}{lll} a_1& a_2 & 1 \\ b_1 &b_2 & 1\\ c_1 &c_2 &1\\ \end{array}\right] ) = picture$$