The area of a triangle and determinants

In summary, the conversation discusses the attempt to prove that the area of a triangle is equal to the determinant of its three points. The person encountered an issue with the signs being incorrect in their proof. They share pictures and ask for clarification on how the two formulas can be identical when one comes from trapezoid subtraction and the other from the determinant. It is clarified that the determinant gives double the area of a counter-clockwise triangle, but in this case the triangle is oriented clockwise, resulting in the opposite area.
  • #1
Yankel
395
0
Dear all,

I was trying to prove that the area of a triangle is equal to the determinant consisting of the three points of the triangle. I got to the end, and something ain't working out. The signs are all wrong.

In the attached pictures I include my proof. Can you please tell me how can the two formulas be identical ? The first is the area coming from trapezoid subtraction , while the second is the determinant.

Thank you !

Clarification: when I say signs are opposite, I mean (y2-y3) vs. (y3-y2) , etc...

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  • #2
Hi Yankel,

The determinant gives double the area of a triangle that is oriented counter-clockwise.
In your case the triangle is oriented clockwise, meaning that we'll find the opposite.
 

1. What is the formula for finding the area of a triangle using determinants?

The formula for finding the area of a triangle using determinants is:
A = 1/2 * |x1 y1 1|
          |x2 y2 1|
          |x3 y3 1|
where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the triangle's vertices.

2. How does using determinants to find the area of a triangle differ from using the traditional formula (1/2 * base * height)?

Using determinants to find the area of a triangle is a more general method that can be applied to any type of triangle, while the traditional formula only works for right triangles. Additionally, the determinant formula does not require the use of a specific "base" and "height" like the traditional formula does.

3. What are the advantages of using determinants to find the area of a triangle?

One advantage of using determinants is that it can be applied to any type of triangle, including equilateral, scalene, and isosceles triangles. It also does not require the use of trigonometric functions, making it simpler and more efficient to use in certain situations.

4. Can determinants be used to find the area of a triangle in 3-dimensional space?

Yes, determinants can be used to find the area of a triangle in 3-dimensional space by adding a third coordinate (z) to each vertex and then using the formula:
A = 1/2 * |x1 y1 z1 1|
          |x2 y2 z2 1|
          |x3 y3 z3 1|
However, in 3-dimensional space, the result will be the area of the triangle's parallelogram, not the triangle itself.

5. How is the determinant of a triangle related to its area?

The absolute value of the determinant of a triangle is equal to twice the area of the triangle. This means that the area of the triangle can be calculated by taking half of the determinant's absolute value.

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