Why to use the normal vector for constructing the plane equation in 3-D Calculus?

Why do people use normal vector for the plane equation in 3-D?


Because in 2-D, we need point and slope, slope is kind of like the vector or the direction of the line, but in 3-D, why do people use normal but not parallel vector? Isn't that only give you the plane equation for the normal vector (which is like the slope) direction instead of the the plane equation that passes through the given points?|||In two dimensions, we can take the normal vector:





%26lt; a , b %26gt;





and turn it into the slope:





- b / a





However, if you have a normal vector in 3D:





%26lt; a , b , c %26gt;





You can't turn it into a fraction - there are three things!


You can't have a "parallel" vector to a plane, since a plane goes many directions. It's also important to remember that the normal vector is more like the slope, whereas the tangent plane is like the tangent line.





Alot of 2D calculus, in high level math, is done with vector methods instead, even though they aren't as necessary.