Find area using vectors (cross product)

2 replies · 3K views · last reply
solina
Messages
1
Reaction score
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.

Green irregular pentagon with red outline and pin markers at its five corners; edge lengths labelled 1.39 m, 1.68 m, 1.7 m, 1.03 m and 1.12 m.


Homework Equations


Determinant formula for the cross product u × v, with a 3×3 matrix whose top row is i, j, k, middle row u1, u2, u3, and bottom row v1, v2, v3.



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.
 
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:
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.

Green pentagon with side lengths labelled 1.39 m, 1.68 m, 1.7 m, 1.03 m and 1.12 m, with a red pin marker at each vertex.


Homework Equations


Determinant-style layout of the cross product u × v: the first row is i, j, k, the second row is u1, u2, u3, and the third row is v1, v2, v3.

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?