Question

Match the parametric equations with the verbal descriptions of the surfaces by putting the letter of the verbal description to the left of the letter of the parametric equation.

1. \mathbf{r} \left( u, v \right) = u \cos v \mathbf{i} + u \sin v \mathbf{j} + u^{2} \mathbf{k}
2. \mathbf{r} \left( u, v \right) = u \mathbf{i} + u \cos v \mathbf{j} + u \sin v \mathbf{k}
3. \mathbf{r} \left( u, v \right) = u \mathbf{i} + \cos v \mathbf{j} + \sin v \mathbf{k}
4. \mathbf{r} \left( u, v \right) = u \mathbf{i} + v \mathbf{j} + \left( 2u - 3v \right) \mathbf{k}


A. circular cylinder
B. circular paraboloid
C. cone
D. plane

Answer 1

1. B

2. C

3. A

4. D

Explanation:

The parametric equations of the circular cylinder are:

`x(u,v)=a\cos v\\y(u,v)=a\sin v\\z(u,v)=u`

If the orientation of the cylinder is changed to have the height
`u` along the x-axis, the parametric equations of the cylinder match:

`3. \mathbf{r} \left( u, v \right) = u \mathbf{i} + \cos v \mathbf{j} + \sin v \mathbf{k}`

The parametric equations of the circular paraboloid are:

`x(u,v)=u\cos v\\y(u,v)=u\sin v\\z(u,v)=u^2`

Using the units vectors the parametric equations match:

`1. \mathbf{r} \left( u, v \right) = u \cos v \mathbf{i} + u \sin v \mathbf{j} + u^(2) \mathbf{k}`

The parametric equations of the cone are:

`x(u,v)=au\cos v\\y(u,v)=au\sin v\\z(u,v)=u`

Using the units vectors and rotating the base of the cone from
`z=0` to
`x=0` the parametric equations match:

`2. \mathbf{r} \left( u, v \right) = u \mathbf{i} + u \cos v \mathbf{j} + u \sin v \mathbf{k}`

The equation left is the equation of a plane:

`4. \mathbf{r} \left( u, v \right) = u \mathbf{i} + v \mathbf{j} + \left( 2u - 3v \right) \mathbf{k}`