Basic Dimension of a set of points question
This is the start of my course and this topic is kind of glossed over in an 'you should already know this' way, but I want to make sure I understand it.
In 3D space, what is the dimension of the set of points that simultaneously satisfy the two equations
(a) x+y=3 and (b) z^2=x^2+y^2
My answer would be that (a) has one dimension because we only need either x or y in order to determine the point. Similarly, (b) needs any 2 of the 3 variables in order to define the missing one, so its dimension is 2? If that is the case, then to simultaneously satisfy both it must have 2 dimensions?
Re: Basic Dimension of a set of points question
Either one of the given two formulas requires two values in order to calculate the third and so are two dimensional but we can use the two equations to solve for one variable. x+ y= 3 is the equation of a plane in three dimensions. is a cone (In cylindrical coordinates it reduces to , two lines at 45 degrees to the xy-plane- think of that rotated around the z-axis). If then so that [itex]x^2- y^= x^2- (3- x)^2= x^2- 9+ 6x- x^2= 9+ 6x= z^2[/itex]. If we let z= t then and . Since x, y, and z can be written in terms of a single variable, it is, by definition of "dimension", one- dimensional.
(The first formula, x+y= 3, has only two variables but that only means that z can be anything. Some points on that plane are (1, 2, 0), (1, 2, 0.5), (0, 3, 20), etc.)