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djverab [1.8K]
3 years ago
6

Match the parametric equations with the verbal descriptions of the surfaces by putting the letter of the verbal description to t

he 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
Mathematics
1 answer:
alukav5142 [94]3 years ago
5 0

Answer:

1. B

2. C

3. A

4. D

Step-by-step 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}

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