Question

Use a computer algebra system to graph the surface represented by the vector- valued function. $$ \begin{array}{l} \mathbf{r}(u, v)=2 \cos v \cos u \mathbf{i}+4 \cos v \sin u \mathbf{j}+\sin v \mathbf{k} \\ 0 \leq u \leq 2 \pi, \quad 0 \leq v \leq 2 \pi \end{array} $$

Short answer

After separating the coordinates from the given vector function and plotting on a computer algebra system, the graph represents an ellipsoid-like shape over the given boundaries.

Step-by-step solution

Understand the Vector-valued function

The given vector function is \( \mathbf{r}(u, v) = 2 \cos v \cos u \mathbf{i} + 4 \cos v \sin u \mathbf{j} + \sin v \mathbf{k} \) with \( 0 \leq u, v \leq 2 \pi \). Where \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) are the unit vectors in the \(x\), \(y\) and \(z\) directions respectively.

Separate the coordinates

Separate into coordinate representations (x, y, and z) from the function \( \mathbf{r}(u, v) \). This will give us: \(x = 2 \cos v \cos u\), \(y = 4 \cos v \sin u\), and \(z = \sin v\).

Plot the Function

Using a computer-based algebra system such as MATLAB or Mathematica, plot the function using the bounds provided in the exercise for \(u\) and \(v\) that is \(0 \leq u, v \leq 2 \pi\).