Is equation of line or curve in 3 space have to be a parametric equation?

[yungman]

I have been confused about this. When come to equation of lines and curves in even 2 space, they automatically go parametric equation x(t), y(t) etc. I just want to verify my understanding:

For 3 space, equation using x, y and z will automatically produce a plane or a surface. In order to produce a line or curve, they have not choice but to represent x, y and z in form of x(t), y(t) and z(t) so it become line or curve.

Therefore a linear line or a curve HAS to be represented by a parametric equation.


Am I correct?

 

[Mark44]

The set of solutions of these two equation defines a curve in R³:
x² + y² = 1
z = 2

 

[yungman]

The set of solutions of these two equation defines a curve in R³:
x² + y² = 1
z = 2

In the calculus books I studied include Sherman Stein and Howard Anton. They only represent lines and curves in 3 space with parametric equation. Also in the chapter of vector value function. All the books I have immediately go to parametric equations.

 

[Mark44]

My point is that curves in space don't have to be represented by parametric equations.

 

[yungman]

Thanks.
Alan

 

[HallsofIvy]

Curves are one dimensional. That means that every point can, theoretically, be determined by using a single number. If you write a curve in terms of parametric equations, then that single number, the parameter, is apparent.

But any time you have n equations in n+1 variables, you can, theoretically, solve for n of the variables in terms of the other one. For example, if you have 3x+ 2y+ z= 4 and x- y- z= 5, adding the two equations eliminates z giving 4x+ y= 9. You can then solve for y in terms of x: y= 9- 4x. Putting that into the second equation, x- (9- 4x)- z= -3x- 9- z= 5 or -z= 14+ 3x so that z= -14- 3x. We could then use x as parameter:
x= t, y= 9- 4t, z= -14- 3t. But the two equations 3x+ 2y+ z= 4 and x- y- z= 5 represent that line (geometrically you can think of the line as the intersection of the two planes given by those equations.

Another way of representing a curve is in what is called the "symmetric form": f(x,y,z)= g(x, y, z)= h(x, y, z). That is just a way of representing the two equations f(x,y,z)= g(x,y,z), g(x,y,z)= h(x,y,z). Again, since there are two equations in three variables, you could, theoretically, solve for two of the variables in terms of the third- its graph is one-dimensional, a curve.

 

[yungman]

Curves are one dimensional. That means that every point can, theoretically, be determined by using a single number. If you write a curve in terms of parametric equations, then that single number, the parameter, is apparent.

But any time you have n equations in n+1 variables, you can, theoretically, solve for n of the variables in terms of the other one. For example, if you have 3x+ 2y+ z= 4 and x- y- z= 5, adding the two equations eliminates z giving 4x+ y= 9. You can then solve for y in terms of x: y= 9- 4x. Putting that into the second equation, x- (9- 4x)- z= -3x- 9- z= 5 or -z= 14+ 3x so that z= -14- 3x. We could then use x as parameter:
x= t, y= 9- 4t, z= -14- 3t. But the two equations 3x+ 2y+ z= 4 and x- y- z= 5 represent that line (geometrically you can think of the line as the intersection of the two planes given by those equations.

Another way of representing a curve is in what is called the "symmetric form": f(x,y,z)= g(x, y, z)= h(x, y, z). That is just a way of representing the two equations f(x,y,z)= g(x,y,z), g(x,y,z)= h(x,y,z). Again, since there are two equations in three variables, you could, theoretically, solve for two of the variables in terms of the third- its graph is one-dimensional, a curve.

Thanks a million, this is exactly what I need, a comparison.

Alan