What is the formula for finding the area of a parallelogram using vectors?

[tnutty]

Homework Statement

Find the area of the parallelogram with the vectices :

A(-2,1)
B(0,4)
C(4,2)
D(-2,1)

Homework Equations

This sections is about vectors, specifically crossproducts.

A = |a|(|b|sin(/theta) = |a x b|

- The length of the cross product a x b is equal to the area of the parallelogram determined
by a and b

The Attempt at a Solution

Instead of using the equation i found the length of each line :

A to B = sqrt(13)
B to C = sqrt(20)
D to C = sqrt(14)
A to D = sqrt(20)

What I did was divide it into 2 triangles find each hypotenuse and multiply it by each other
then divide it by 2. I don't know if there is a formula similar to this but the answer I got
was 16.748.

The problem I am having is to use the definition in the Relevant equation sections above.
How do I know which points would be a vector. If I know that then I can
probably figure about there Cross product and find its magnitude which is the parallelogram area.

 

[VeeEight]

You can draw the points in the plane and then make vectors out of them to apply the cross product formula. See here: http://www.jtaylor1142001.net/calcjat/Solutions/VCrossProduct/VCPAParallelogram.htm

 

[tnutty]

Oh I see, All I need is x and y : x*y = Area, where x & y are vectors.

But will it always be the case that the length of the A to D will be equal to the length B to C?

 

[VeeEight]

Remember to take the magnitude of the cross product (i.e |x*y| )

No.

 

[tnutty]

How do I counteract that if it does not have the same length on the opposite side.

Do I have to do 2 cross product?

 

[HallsofIvy]

Opposite sides in a parallelogram are always of equal length.