SPATIAL PYTHAGOREAN THEOREM IN TRI-RECTANGULAR TRIHEDRON VIA GRAM MATRIX

Abstract

The demonstration of this work uses vector calculus and algebra topics to prove the generalized Pythagorean theorem in space using a tri-rectangular trihedron that gives rise to a tetrahedron. The purpose of this paper is to demonstrate that the sum of the squares of the areas of the faces of a tetrahedron is equal to the square of the opposite face. Some concepts of scalar product, Lagrange identity and Gram matrix were widely used. The results showed that its various applications lead to a number of important results and conclusions in Mathematics, Engineering, Science and Geometry. With this, it is concluded that the generalization of the Pythagorean theorem by tri-rectangle trihedron using the Gram matrix and this Important Lagrange Identity, is an important contribution, as it expands its relevance and importance in mathematics and other areas of knowledge.