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

In summary, to find the area of a parallelogram with given vertices, you need to find two vectors that share a common point. These vectors can be found by taking the difference of the coordinates of two points. Once you have these two vectors, you can calculate the magnitude of their cross product to find the area of the parallelogram. No visualization is necessary, just pick any three of the four given points and use them to find the two vectors. The area of the parallelogram with the given vertices is approximately 5*sqrt(30).
  • #1
Townsend
232
0
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? :bugeye:
 
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  • #2
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).
 
  • #3
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.
 
  • #4
LeonhardEuler said:
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
 

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

What is a cross product?

A cross product is a mathematical operation that takes two vectors and produces a third vector that is perpendicular to both of the original vectors. It is commonly used in physics and engineering to calculate torque, angular momentum, and magnetic fields.

How is a cross product calculated?

To calculate the cross product of two vectors, you first take the determinant of a 3x3 matrix using the components of the two vectors. The resulting vector will have a magnitude equal to the area of the parallelogram formed by the two original vectors, and its direction will be perpendicular to both vectors according to the right-hand rule.

What is the significance of the cross product?

The cross product is important in many fields because it allows us to determine the direction of a vector that is perpendicular to two other vectors. This is useful in determining the direction of forces, rotations, and magnetic fields in three-dimensional space.

Can the cross product be used with more than two vectors?

Yes, the cross product can be extended to calculate the vector perpendicular to multiple other vectors. This is known as the triple product and is used in advanced physics and engineering calculations.

What are some real-life applications of the cross product?

The cross product has many practical applications, such as in calculating the torque required to turn a wrench, the force exerted by a rotating object, and the direction of a magnetic field created by a current-carrying wire. It is also used in computer graphics to determine the orientation of objects in 3D space.

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