Area of a parallelogram using cross product, how can length be equal to area?

Question

We get a vector by a cross product and its length (magnitude) is the area of the parallelogram. How is this possible as the unit of length is meters and unit of area is meters squared?

Answer 1

If the two vectors are $(a,b)$ and $(c,d)$, then the magnitude of their cross product is $ad - bc$. So, if $a,b,c,d$ all have dimensionality "length", then $ad$ and $bc$ both have dimensionality $\text{length}^2$, and so does the cross product.

If $a,b,c,d$ are expressed in meters, the cross product formula will give you an area in square meters.