The problem requires eliminating the parameters $u$ and $v$ from the given vector-valued function to find the corresponding rectangular equation for the surface. Let's analyze it step by step.

Given:

$\mathbf{r}(u, v) = u \mathbf{i} + v \mathbf{j} + \frac{v}{2} \mathbf{k}$ This can be expressed in components: $x = u, \quad y = v, \quad z = \frac{v}{2}.$

Step 1: Solve for $u$ and $v$ in terms of $x, y, z$:

From $x = u$, we directly have: $u = x.$

From $y = v$, we directly have: $v = y.$

From $z = \frac{v}{2}$, substituting $v = y$, we get: $z = \frac{y}{2}.$

Step 2: Eliminate parameters:

From $z = \frac{y}{2}$, multiply both sides by 2: $y = 2z.$

Final Equation:

The rectangular equation of the surface is: $y = 2z.$

Step 3: Identify the Surface:

The equation $y = 2z$ represents a plane in three-dimensional space.

Answer:

• Rectangular Equation: $y = 2z$
• Surface: Plane

Would you like more details on this process? Let me know!

Related Questions:

1. What are vector-valued functions, and how are they used to represent surfaces?
2. How can you distinguish between different surfaces (plane, cone, sphere, etc.) based on their equations?
3. What is the geometric interpretation of the parameter $u$ in this example?
4. How would the equation change if $z = \frac{v}{3}$ instead of $z = \frac{v}{2}$?
5. Can this method of parameter elimination be applied to more complex vector-valued functions?

Tip:

When working with vector-valued functions, remember to carefully map the parameter relationships to the coordinate axes to simplify your solution.