Step 1

In polar coordinates, \(x=r\cos\theta\), \(y=r\sin\theta\)

Using this rule, using appropriate substitutions:

\(\displaystyle{x}={2}{\cos{{u}}}\), \(\displaystyle{y}={4}{\sin{{u}}}\)

Step 2

Considering \(\displaystyle{z}={v}\), the vector form is given by:

\(\displaystyle{r}{\left({u},{v}\right)}={2}{\cos{{u}}}{i}+{4}{\sin{{u}}}{j}+{v}{k}\)

Step 3

Answer:

\(\displaystyle{r}{\left({u},{v}\right)}={2}{\cos{{u}}}{i}+{4}{\sin{{u}}}{j}+{v}{k}\)