To find the parametric equation $\vec{r}(t)$ of the curve $C$, which is the intersection of the two surfaces defined by:

1. $x^2 + y^2 = 4$ (a cylinder),
2. $x + y + z = 1$ (a plane),

we can proceed as follows:

Step 1: Parameterize the Cylinder Equation

The cylinder equation $x^2 + y^2 = 4$ is a circle of radius 2 in the $xy$-plane. A standard parameterization for this circle is: $x = 2 \cos t, \quad y = 2 \sin t$ where $t$ is the parameter that goes from $0$ to $2\pi$ to cover the entire circle.

Step 2: Express $z$ in Terms of $x$ and $y$ Using the Plane Equation

From the plane equation $x + y + z = 1$, we can solve for $z$ in terms of $x$ and $y$: $z = 1 - x - y$ Substitute $x = 2 \cos t$ and $y = 2 \sin t$ from our parameterization: $z = 1 - (2 \cos t) - (2 \sin t)$

Step 3: Write the Parametric Equation

Now we have the parametric equations for $x$, $y$, and $z$ in terms of $t$: $\vec{r}(t) = \langle x, y, z \rangle = \langle 2 \cos t, 2 \sin t, 1 - 2 \cos t - 2 \sin t \rangle$ where $t$ ranges from $0$ to $2\pi$ to cover the entire curve.

Final Answer

The parametric equation of the curve $C$ is: $\vec{r}(t) = \langle 2 \cos t, 2 \sin t, 1 - 2 \cos t - 2 \sin t \rangle, \quad t \in [0, 2\pi]$

Would you like a further breakdown of each step, or do you have any questions?

Here are some related questions to expand on this concept:

1. How would you find the tangent vector of $\vec{r}(t)$ at a specific point?
2. What is the geometric interpretation of this parametric curve?
3. How does the orientation (counterclockwise) affect the parametric equations?
4. How would the parameterization change if the cylinder had a different radius?
5. Can we find the arc length of this curve over one complete cycle?

Tip: When working with curves defined by intersections, always start by parameterizing the simpler surface and then substitute it into the other equation.