MHB The area of a triangle and determinants

Click For Summary
The discussion centers on proving that the area of a triangle can be represented by a determinant formed by its three vertices. The original poster encounters issues with sign discrepancies in their proof, specifically regarding the orientation of the triangle. It is clarified that the determinant yields double the area for a counter-clockwise orientation, while a clockwise orientation results in a negative value. This indicates that the signs in the formulas must be adjusted based on the triangle's orientation. Understanding these orientation effects is crucial for aligning the two area representations.
Yankel
Messages
390
Reaction score
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...

View attachment 7656

View attachment 7657

View attachment 7658

View attachment 7659
 

Attachments

  • 1.PNG
    1.PNG
    28.4 KB · Views: 114
  • 2.PNG
    2.PNG
    17.7 KB · Views: 126
  • 3.PNG
    3.PNG
    4.9 KB · Views: 110
  • 4.PNG
    4.PNG
    7.1 KB · Views: 122
Mathematics news on Phys.org
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.
 

Thread 'Significant figures for inherently bound values'

I have been insisting to my statistics students that for probabilities, the rule is the number of significant figures is the number of digits past the leading zeros or leading nines. For example to give 4 significant figures for a probability: 0.000001234 and 0.99999991234 are the correct number of decimal places. That way the complementary probability can also be given to the same significant figures ( 0.999998766 and 0.00000008766 respectively). More generally if you have a value that...
View full post »

Similar threads

MHB What is the area of triangle $STV$?
  • · Replies 1 ·
Replies
1
Views
1K
MHB Area of Triangle Shaded Region
  • · Replies 5 ·
Replies
5
Views
2K
MHB Prove Area of Equilateral Triangle
  • · Replies 9 ·
Replies
9
Views
2K
MHB Calculate Area of Triangle ABC
  • · Replies 2 ·
Replies
2
Views
5K
Comp Sci Computing the Length & Area of a 3D Triangle
  • · Replies 1 ·
Replies
1
Views
2K
MHB Equilateral triangle within a circumscribed circle
  • · Replies 5 ·
Replies
5
Views
3K
MHB Prove Equal Area Triangle & Parallelogram, $\angle ADC=90^{\circ}$
  • · Replies 9 ·
Replies
9
Views
3K
MHB How Can You Calculate the Area of a Polygon Using Coordinates?
  • · Replies 3 ·
Replies
3
Views
2K
Find the third coordinate of a vertex of an equilateral triangle
  • · Replies 11 ·
Replies
11
Views
10K
MHB Maximize Trapezoid Area with 3 Equal Sides | Leprofece Answer
  • · Replies 1 ·
Replies
1
Views
3K