Cuboid points coordinates in 3D I have this gray cuboid, bounded by the white cube and I need to draw a line (the red line, which passes form the center of the edge, the center of the cuboid, to the other edge) The info I have are the coordinates of the center, the orientation of the cuboid, and the position of the 2 corners of the cube (the 2 points that the green line passes between) ![enter image description here]( Is it possible to do so with the current info?

To sum up the discussion in the notes: The OP knows that the cuboid shares the same center as the cube and has six points of contact with the cube, and also knows the orientation of the cuboid. Also, the OP knows the equations of the planes of the faces of the cube. Therefore, the solution method is to let $l,w,h$ be the unknown dimensions of the gray cuboid. Then based on its orientation, you can write an expression in $l,w,h$ for the the "top" vertex of the cuboid. Since that touches the top face of the cube, you know it must satisfy the equation of that plane, which yields one equation in $l,w,h$. Do the same for the two other independent contact points (i.e., not the "bottom" vertex, which by symmetry will give you an equation equivalent to the first), and you will have three equations in $l,w,h$ which you solve, and then you know everything about the cuboid.