Answer to Question #314912 in Math for math student

Question

Given the following points: 𝑃0 = (−1, −1,3),𝑃1 = (−1,3, −1),𝑃2 = (3,5,3)𝑃3 = (3,3,5), 𝑃4 = (− 1 2 , 3,3). a) Show that 𝑃0, 𝑃1,𝑃2, 𝑃3 lie on the same plane, H, and find the implicit equation of H. A pyramid is defined by the plane H and the following triangular faces: (𝑃0,𝑃2,𝑃4 ), (𝑃0, 𝑃1,𝑃4 ), (𝑃1,𝑃3,𝑃4 ), (𝑃2,𝑃3, 𝑃4 )

b) Determine the outwards facing unit normal vector of each triangular face. c) Calculate the implicit representation of the planes containing each face of the shape. d) For each of the following points determine if it is inside or outside the shape (hint: the point is inside the shape if it lies on the same side for all the planes) i) (− 1 2 , 1,2) ii) (1,0,1) iii) (3,2,4)