Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function $\mathbf { r } ( u , v ) = 2 \cos v \cos u \mathbf { i } + 2 \cos v \sin u \mathbf { j } + 3 \sin v \mathbf { k }$ and sketch the graph.

Question

Quizplus · Accepted Answer

We can eliminate the parameters using the x, y, and z components of the vector-valued function:

$$x=2\cos v \cos u $$

$$y=2\cos v \sin u $$

$$z= 3 \sin v $$

Squaring the first two equations and adding them gives:

$$x^2+y^2=4\cos^2 v$$

Multiplying the third equation by $\frac{1}{3}$ and taking the square of each equation gives:

$$\frac{1}{9}z^2=\frac{1}{9}\cdot 9 \sin^2 v=\sin^2 v$$

Now, we can eliminate $v$ by taking the square root of both sides and combining the results:

$$x^2+y^2=4\cos^2 v$$

$$\frac{1}{9}z^2=\sin^2 v$$

$$\frac{x^2}{4\cos^2 v}+\frac{y^2}{4\cos^2 v}+\frac{z^2}{9}=1$$

Therefore, the rectangular equation for the surface is:

$$\frac{x^2}{4\cos^2 v}+\frac{y^2}{4\cos^2 v}+\frac{z^2}{9}=1$$

To sketch the graph, we note that $0\leq v \leq \pi$ and $0\leq u \leq 2\pi$, which means the surface is a closed ellipsoid with its center at the origin, axis lengths of $2/\sqrt{3}$ in the $xy$-plane and $3$ along the $z$-axis.

Find the Rectangular Equation for the Surface by Eliminating the Parameters