why can't there be a cartesian equation of a plane in 3d-space?
Greetings Shirel:
Was this claim put forward to you by your classroom math teacher? I ask this because, while approaching with some uncertainty, the terminology, as I understand it, goes counter to your assertion. First, by "Cartesian Equation", I assume your reference to imply an equation in two variables, typically x and y, whose solutions delineate a set of ordered pairs, (x,y), existing in one-to-one correspondence with a set of points in the plane that defines the "graphical representation" of such equation. But existence of a graph in 2-space does not pre-empt existence of a second function mapping solutions to some other image set and vice versa. That is, function f can map a subset of 2-space (ordered pair solutions) onto a sedcond subset of 2-space (points in the plane), while function g maps the same "from set" into 3-space. And, provided f and g are one-to-one functions, graphical representations of like sets are unique. Let's swap the technical lingo for plain English as we consider an example or two. The graph of x=4 exists as a single point on the real line. It likewise exists as an infinitely numbered set of points in the plane because the equation x=4 places no restrictions on y. Hence (4,-5),(4,0), indeed (4,r) for all real numbers, r, all qualify as solutions. Now, consider the "Cartesian Equation" y=mx+b whose graph is traditionally thought to exist in the plane. But the absence of any constraints opon z qualifies all ordered triples (x, mx+b, z) as solutions of the equation. This latter statement not only serves to refute the initial claim, but rather appears to negate the claim entirely. That is, 3-space, in fact, contains an infinite supply of planes whose equations are "Cartesian" by definition. Moreover, every line in two-space corresponds to a "Cartesian equation" whose graph is a planar subset of 3-space. In truth, a line in the xy plane is just the intersection (trace) of two planes -- the xy plane with some other non-parallel plane. We know this to be true according to the geometric postulate asserting the intersection of two unique planes as a single line. ...and so it goes and so it goes, and you're the only who knows (-B.Joel).
Regards,
Rich B.