Find a rectangular equation for the surface

[stevecallaway]

Homework Statement

r(u,v)=u i +v j +(1/2)v k

Homework Equations

The Attempt at a Solution

x=u : y=v : z=(1/2)v
because x=u and y=v, x & y are the parameters
so r(x,y)=x+y+(1/2)y=x+(3/2)y
but the answer says it is y-2z=0. What am I not seeing correctly?

 

[LCKurtz]

Homework Statement

r(u,v)=u i +v j +(1/2)v k

x=u : y=v : z=(1/2)v
because x=u and y=v, x & y are the parameters
so r(x,y)=x+y+(1/2)y=x+(3/2)y

But r is given as a vector, so this last equation makes no sense. What you want to do is eliminate the u,v,w variables as much as possible. Do you see a relation between y and z? And it looks like y and z don't depend on x...

 

[stevecallaway]

I think I see a relation between y and z and that is y=2z. So is that all that I'm supposed to do is find a relationship from among the original equation and have that equal to zero? Because y-2z=0 is supposed to be the answer, but what happens to the u i?

 

[LCKurtz]

The vector parametric form is one way to write the equation of a surface. An equation of the form f(x,y,z)=0 is another way. The parametric way is written as a vector function and the other way as a scalar equation. Your vector representation is equivalent to your three equations: x=u, y=v, z=(1/2)v. In this case there is the relation y = 2z which is independent of x, which can be anything. You would normally write the equation y = 2z. The other variable, which is now missing, can be anything. This is characteristic of a cylindrical surface -- it is just the plane formed by taking the line y = 2z in the zy plane and extending or "sweeping" it in the x direction.

[Edit] typos