Cross Products relation to area.

In summary, the area of a parallelogram can be calculated by finding the cross product of two vectors, v1 and v2, and taking the magnitude of the resultant vector, which is equal to the product of the magnitudes of v1 and v2 and the sine of the angle between them. This can be visualized by imagining the parallelogram formed by v1 and v2 and using the cross product to find the height of the parallelogram, which can then be multiplied by the base to find the area.
  • #1
mkkrnfoo85
50
0
"The area of a parallelogram spanned by two vectors, v1 and v2, is ||v1 X v2||."

Would someone help me understand why this is true?
 
Physics news on Phys.org
  • #2
Imagine a paralllelogram, and imagine 2 vectors V1, and V2, which have a Theta angle between them. Now You know the magnitude or modulus of the resultant vector of the cross product is given by [itex] |V_{1}||V_{2}| \sin{\theta} [/itex], imagine V2 is a horizontal vector (only a component of x), so it will be a side of the parallelogram (the base), now imagine V1 is directed at a theta angle, if you do V1sin theta, you will get the height of the parallelogram, so base times height equals area.
 
  • #3
ah. That makes a lot of sense. Thanks
(I kinda forgot about |v1||v2|sin(theta).. The things I don't see... :frown:)
 

What is a cross product?

A cross product is a mathematical operation that takes two vectors as inputs and produces a third vector that is perpendicular to both of the input vectors.

How is the cross product related to area?

The magnitude of the cross product of two vectors is equal to the area of the parallelogram formed by those two vectors. This means that the cross product can be used to calculate the area of a 2D shape.

What is the formula for calculating the cross product?

The formula for calculating the cross product of two 3D vectors (a and b) is: a x b = (aybz - azby)i + (azbx - axbz)j + (axby - aybx)k

Can the cross product be used for any shape?

The cross product can only be used to calculate the area of 2D shapes. It cannot be used for 3D shapes.

How is the direction of the cross product determined?

The direction of the cross product is determined by the right-hand rule. This means that if you curl the fingers of your right hand from vector a to vector b, your thumb will point in the direction of the resulting cross product vector.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
877
  • Mechanical Engineering
Replies
22
Views
423
Replies
6
Views
499
  • Special and General Relativity
Replies
10
Views
1K
Replies
2
Views
1K
Replies
2
Views
227
Replies
16
Views
4K
Replies
3
Views
286
  • Calculus
Replies
4
Views
1K
  • Calculus
Replies
1
Views
994
Back
Top