Area of Parallelogram

Recall that the definition for cross product is 
 .If we find the magnitude of the cross product, we get 
because 

1) n is a unit vector and therefore |n| = 1, and

2)  for .

Now, lets take a look at a parallelogram that is formed by two vectors u and v.

The area of the parallelogram is 

area = base * height

which the area is equals to the magnitude of . 

As such we can use the cross product to find area of parallelogram formed by vectors u and v:

The graph below shows the parallelogram in space. The calculation at the top left of the graph is the calculation using the cross product, whereas the right uses the previous area formula. Try to move point A and B to change the shape of parallelogram and observe the area calculted using both methods.

Betty, Created with GeoGebra and Thomas Calculus, Pearson.