Let the vectors q represent the two adjacent sides of a parallelogram with one of its vertices at the origin.

The other three vertices are:

(a,b), (c,d) and (a + c, b + d).

p = ai + bj

q = ci + dj

Representing these as 2 rows of a matrix,

The determinant of 2 × 2 matrix gives the area of the parallelogram.

We have

= a₁i + b₁j + c₁k = a₃i + b₃j + c₃k