Given 3 points $x_1,x_2,x_3\in\mathbb{R}^n$ in Euclidean space. What is the area of the triangle they span?
Of course the formulas for $\mathbb{R}^2$ and $\mathbb{R}^3$ are well known. But how do the formulas generalize to higher dimensions?
My motivation is to generalize the formula stated in the publication Polygon Laplacian Made Simple by Bunge et al. Please, note that I am not looking for the volume of a "simplex". In my problem, the number of points is always 3 regardless of the dimension. Thank you!
Best Answer
Hint: You can use Heron's formula Remember that the distance between two points $x,y$ in euclidean standard space is: $$d(x,y)=\sqrt{\sum_{i=1}^n (x_i-y_i)^2}$$