How do I find the area of a parallelogram using cross product?

[Townsend]

11. Calculate the area of the parallelogram having the vertices
(1,2,3),(4,-2,1),(-3,1,0), and (0,-3,-2).

To solve this problem I need to find two vectors that share a common point? Then I can take the magnitude of the cross product of those two vectors to find the area of the parallelogram...no problem.

The problem is how do I know what two vectors will have a common point? Am I just suppose to think in 3d and see it?

 

[LeonhardEuler]

Just find two such vectors by saying that one goes from point A to point B while the other goes from point A to point C. The components of these vectors can be computed by (B-A) and (C-A).

 

[Physics Monkey]

No visualization is necessary. Pick any three of the four given points. We will call them p1, p2, and p3. Let one of those three be your base point (the common point). Now form the difference v1 = p2 - p1 and v2 = p3 - p1 where p1 is your base point. The two vectors v1 and v2 are the vectors you are looking for, they are the sides of the parallelogram.

 

[Townsend]

Just find two such vectors by saying that one goes from point A to point B while the other goes from point A to point C. The components of these vectors can be computed by (B-A) and (C-A).

Ah! I see... no problem...so it's what 5*sqrt(30) then...

thanks