Write an equation for a surface which contain the curves defined by following vector function
$r(t)=\bigg<3t,e^{t},1-t^2\bigg>$
What i try:
Campare with $r(t)=\bigg<x,y,z\bigg>$
We get $x=3t,y=e^{t},z=1-t^2$
$$9z=9-(3t)^2=9-x^2\Longrightarrow 9z=1-x^2$$
I did not understand what is the use of $y$ coordinate Here. Help me please . Thanks
There is no need to use the $y$ coordinate. The equation you found describes a set which obviously contains the curve. Now why is it a surface ? It depends on what you definition of a surface is but intuitively, what can we say? In the plane $y=y_0$, the equation describes a parabola. Since this parabola is the same whatever the value of $y_0$ is, the surface is the union of all the possible translations of this parabola along the $y$ axis, and as a result it is indeed a surface. Does that answer your question?
Roughly speaking (I say that because there are some degenerate case in which this is not true), a subset of $\mathbb R^n$ described by $k \in [\![0,n]\!]$ non-redundant scalar equations is of dimension $n-k$. Here, you are in dimension 3 and your set is described by one scalar equation so it is of dimension 2 and it is a surface. Note that the notion of 'dimension' that I refer to here is a more general notion than the dimension of vector spaces. The concept that is behind it is differentiable manifold, but it is a somewhat advanced domain.
A surface in 3D can be given by an equation, while a curve is given by two independent equations.
Your curve satisfies the equations $9z=9-x^2$ and $y=e^{x/3}$. The first equation alone gives you a surface which contains the curve. The fact that the generating equation for this surface does not contain $y$ just means that the surface has a translational symmetry in the $y$ direction. This is similar to the cylinder $x^2+y^2=1$ which has a translational symmetry in the $z$ direction.
It may seem odd that you did not have to know $y(t)$, but this is just a manifestation of the fact that there are many surfaces containing this curve. The full parameterization of the curve contains extra information which we don't have to use: any two coordinates could have sufficed in this case.
In fact, there is slightly different way to see this curve as the intersection curve of two surfaces.
Indeed, by elimination of $t$ between equation (a) and equation (c), we obtain relationship (A), whereas elimination of $t$ between equation (a) and equation (b) gives relationship (B) below
$$\begin{cases}z&=&1−(x^2)/9& \ \ \ (A)\\ y&=&e^{x/3}& \ \ \ (B)\end{cases} \tag{1}$$
Otherwise said the curve is the intersection curve of surfaces with equations (A) and (B) which are (generalized) cylinders as shown on the figure below.
Therefore, the conclusion is that either (1)(A) or (1)(B) provides an answer to your question.
More generally, if we consider equations (1)(A) and (1)(B) written under the form
$$\begin{cases}z-1+(x^2)/9&=&0& \ \ \ (A)\\ y-e^{x/3}&=&0& \ \ \ (B)\end{cases}\tag{2}$$
any linear combination of these equations, for example :
$$3(z-1+(x^2)/9)+4(y-e^{x/3})=0$$
is also a solution (it will correspond to a surface which is "in-between" the two represented surfaces).
