1. ## Parametric Surfaces

Find the rectangular equation for the surface given by $\displaystyle r(u,v)=\frac{v}{2}i+uj+vk$ by eliminating the paranmeter.

I'm having a bad day. I can see that $\displaystyle 2x=z$. But, that's as far as I get. Any Help?

2. Originally Posted by VonNemo19
Find the rectangular equation for the surface given by $\displaystyle r(u,v)=\frac{v}{2}i+uj+vk$ by eliminating the parameter.

I'm having a bad day. I can see that $\displaystyle 2x=z$. But, that's as far as I get. Any Help?

As it turns out, $\displaystyle 2x=z$ is the equation of the plane.

The position vector, $\displaystyle \displaystyle \vec r(u,v)=\frac{v}{2}\,\hat i+u\hat j+v\hat k,$ gives the coordinates of a point at $\displaystyle \displaystyle (x,\ y,\ z)$$\displaystyle \displaystyle =\left(\frac{v}{2},\ u,\ v\right)\,. \displaystyle \displaystyle \vec r(0,0)=0\hat i+0\hat j+0\hat k, so the surface passes through the origin, (0, 0, 0). Define vector, \displaystyle \displaystyle \vec A$$\displaystyle \displaystyle =\vec{r}(2,\ 0)-\vec{r}(0,\ 0)=1\hat i+0\hat j+2\hat k$.

Define vector, $\displaystyle \displaystyle \vec B$$\displaystyle \displaystyle =\vec{r}(0,\ 1)-\vec{r}(0,\ 0)=0\hat i+1\hat j+0\hat k$.

Notice that any vector of the form $\displaystyle \displaystyle \vec r(u,v)=\frac{v}{2}\,\hat i+u\hat j+v\hat k,$ may be written as a linear

combination of the constant vectors, $\displaystyle \displaystyle \vec A$ and $\displaystyle \displaystyle \vec B$.

Therefore, the surface specified by $\displaystyle \displaystyle \vec r(u,v)$ is a plane.

Vectors, $\displaystyle \displaystyle \vec A$ and $\displaystyle \displaystyle \vec B$ lie in the plane, so their vector product (cross product) is perpendicular to the plane.

$\displaystyle \displaystyle \vec A\times\vec B = \begin{vmatrix} \ \hat i&\hat j&\hat k\ \\ \ 1&0&2\ \\ \ 0&1&0\ \end{vmatrix}=-2\,\hat i+0\,\hat j+1\,\hat k$

So, the equation of the plane is $\displaystyle \displaystyle -2(x-0)+0(y-0)+1(z-0)=0,$

which may be written as $\displaystyle \displaystyle 2x-z=0\,.$

3. Hey man, thanks a lot.