If a positional vector in a 3 dimensional plane is given by (x,y,z), what would be the tangent of the angle the vector makes with the x axis ? Please give an intuitive explaination, supplemented with diagrams, if possible.

if you are given three points, you can take the cross product of the vectors between two pairs of points to determine a normal vector n. Pick one of the three points, and let a be the vector representing that point. Then, the same equation described above, n⋅(x−a)=0.