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As we all know that area of a triangle is given by absolute value of the determinant of this matrix$ A = \dfrac 1 2 {\left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix} \right|}, $ for 2D coordinates system. However is there a formula involving determinants for 3D Coordinates.
(Note: I know that we can use cross products of vectors to find area of triangle for 3D coordinates and that it works for general n coordinates but I wanted to know if there exists a formula for 3D in determinant form(even if its probably impractical to use it))

Sebastiano
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mrtechtroid
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  • A cross product can be expressed as a determinant, can it not? – Théophile Feb 03 '23 at 13:49
  • @Théophile firstly while taking cross products you need to do $x_2-x_1$ and similar stuff and then you need to find the magnitude of the cross product. but you cannot find the magnitude of the cross product directly using determinants and instead need to do a intermediate step of finding the vector and then finding its magnitude using square roots of sum of square of its components. – mrtechtroid Feb 03 '23 at 13:54
  • You might be interested in the answer to this question: https://math.stackexchange.com/questions/4437123/area-of-a-parallelogram-spanned-by-two-4d-vectors-without-using-trigonometry/4437528#4437528 – awkward Feb 03 '23 at 15:08
  • Yes, the volume of simplices can be expressed by a generalization of this determinant. – Lutz Lehmann Feb 03 '23 at 15:39
  • Please see https://math.stackexchange.com/questions/128991/how-to-calculate-the-area-of-a-3d-triangle – van der Wolf Feb 03 '23 at 15:59

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