For any plane in any number of integer Euclidean dimensions we have if p1, p2, and p3 are not all collinear that p1+a(p2-p1)+b(p3-p1) defines that plane. a and b are just numbers and this holds for whole points and single coordinates. For a line we have for l1 and l2, l1(1-c)+l2(c) where c is a single number and l1 and l2 are either whole points or coordinates of their points.
This allows us to solve for plane and line intersections. p1+a(p2-p1)+b(p3-p1)=l1(1-c)+l2(c) is an equation to use. This supplies us with sets of equations that we can solve using linear algebra and a matrix of 4 times number of dimensions. Even less if we can hold something constant or rotate so it is like constant. With this we define for any 3 not all collinear or with any equal points, a plane, and for any 2 not equal points, a line, and can define their properties in many dimensional space. We can also use law of cosines with distances on our plane because it is like 2d and define perpendicular directions if we want. Recommended for if things get close to equal.