To find the vector parametric equation for the circle with radius 3, center at $(1, 4, 3)$, and lying in the plane $y = 4$, we need to follow these steps:

1. Circle Characteristics

• The center of the circle is $(1, 4, 3)$.
• The radius of the circle is $3$.
• The circle lies in the plane $y = 4$, which means the $y$-coordinate remains constant.
• We need the circle to rotate counterclockwise when viewed from the origin. In the $y = 4$ plane, this means it will rotate in the $xz$-plane.

2. Parametric Form

We know that a circle with radius $r$ can be parametrized in the $xz$-plane (since $y$ is constant) as: $\langle x(t), y(t), z(t) \rangle = \langle r \cos(t), 0, r \sin(t) \rangle$ for $t \in [0, 2\pi]$.

3. Adjust for Center and Radius

To adjust this for the center $(1, 4, 3)$ and radius $3$:

• The radius remains $3$, so the parametric equations become: $\langle x(t), y(t), z(t) \rangle = \langle 1 + 3 \cos(t), 4, 3 + 3 \sin(t) \rangle$

This equation gives the parametric form of the circle. As $t$ varies from $0$ to $2\pi$, it traces the circle of radius $3$ with center $(1, 4, 3)$ in the plane $y = 4$, rotating counterclockwise when viewed from the origin.

Final Parametric Equation:

$\mathbf{r}(t) = \langle 1 + 3 \cos(t), 4, 3 + 3 \sin(t) \rangle$ for $t \in [0, 2\pi]$.

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Would you like further clarification on any part of the process or additional details?

Here are five related questions for deeper understanding:

1. How would the parametric equation change if the circle were rotated clockwise?
2. What is the significance of the plane $y = 4$ in this problem?
3. How would you modify the parametric equation for a circle in a different plane, like $z = 3$?
4. How can you verify that the curve is indeed a circle with radius 3?
5. What would the parametric equation be for an ellipse instead of a circle?

Tip: When working with parametric equations, visualizing the movement of each coordinate can help confirm the correctness of your equation.