Find area using vectors (cross product)

In summary, the conversation discusses how to calculate the area of a polygon using the cross product. The person must divide the polygon into triangles and establish coordinates for each vector. However, there is some confusion about the correct coordinates and whether there is enough information to solve the problem.
  • #1
solina
1
0

Homework Statement


Hello, I've been trying to solve this problem, but in the examples that my teacher gave me didn't include something like this, I know how to calculate area but only if I have all the coordinates established.
I need to find the area using the cross product.

Untitled.jpg


Homework Equations


a92ba9287031332f696b2ae85382e1aa.png



The Attempt at a Solution


I know I must divide the polygon into triangles, so I can calculate the area.
I need a starting point for the coordinates.
For example I can call the starting point A.
A(0,0)
Then I need more coordinates in order to establish each vector.
B(0,1.03)
C(0,1.12)

After that...
AB = 0i + 1.03j
AC = 0i + 1.12j
CB = 0i + 2.15j

A=[itex]\frac{1}{2}[/itex](AB x AC)

If what I wrote above is right then I think I can solve It, but I'm almost absolutely sure that is wrong.
I need to know how to declare each vector so I can calculate the area.
I hope is clear enough, and thanks in advance for taking your time to read and help me with this problem.
 
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  • #2
Do you know geometrically what area the cross product represents?

Next you need the identify the points on the diagram.

I assume you meant point A to be the one in the lower left
and that the x-axis is along the lower left boundary line, correct?

If so then the vectors you've written are wrong:

AB = <1.03, 0>

And

AC = <0, 1.14>

...
 
Last edited:
  • #3
solina said:

Homework Statement


Hello, I've been trying to solve this problem, but in the examples that my teacher gave me didn't include something like this, I know how to calculate area but only if I have all the coordinates established.
I need to find the area using the cross product.

Untitled.jpg


Homework Equations


a92ba9287031332f696b2ae85382e1aa.png

The Attempt at a Solution


I know I must divide the polygon into triangles, so I can calculate the area.
I need a starting point for the coordinates.
For example I can call the starting point A.
A(0,0)
Then I need more coordinates in order to establish each vector.
B(0,1.03)
C(0,1.12)

After that...
AB = 0i + 1.03j
AC = 0i + 1.12j
CB = 0i + 2.15j

A=[itex]\frac{1}{2}[/itex](AB x AC)

That isn't correct without absolute value signs. The area is numerically equal to 1/2 the length of the cross product.

If what I wrote above is right then I think I can solve It, but I'm almost absolutely sure that is wrong.
I need to know how to declare each vector so I can calculate the area.
I hope is clear enough, and thanks in advance for taking your time to read and help me with this problem.

Offhand, I don't think your problem has a solution. I don't think knowing the lengths of the sides of a pentagon is enough information to determine its area. Is there more information you haven't included?
 

FAQ: Find area using vectors (cross product)

1. What is the cross product and how is it used to find area?

The 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 used to find the area of a parallelogram formed by the two vectors, by taking the magnitude of the cross product vector.

2. How do you calculate the cross product of two vectors?

To calculate the cross product of two vectors, you first need to find the determinant of a 3x3 matrix with the first row being the unit vectors i, j, and k, the second row being the components of the first vector, and the third row being the components of the second vector. Then, take the absolute value of the determinant to get the magnitude of the cross product vector.

3. Can the cross product be used to find the area of any shape?

No, the cross product can only be used to find the area of a parallelogram formed by two vectors. It cannot be used to find the area of other shapes such as circles or triangles.

4. What is the difference between the cross product and dot product?

The cross product produces a vector as its result, while the dot product produces a scalar. The cross product also gives information about the direction of the resulting vector, while the dot product does not.

5. Can the cross product be used in 3-dimensional space?

Yes, the cross product can be used in 3-dimensional space to find the area of a parallelogram formed by two 3-dimensional vectors. The resulting vector will also be a 3-dimensional vector.

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