Find a vector parametric equation for the circle of radius 3 with center 
(1, 4, 3)
 in the 
y = 4
 plane that rotates counterclockwise when viewed from the origin. (Your instructors prefer angle bracket notation < > for vectors.)

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.

Find a vector parametric equation for the circle of radius 3 with center (1, 4, 3) in the y = 4 plane that rotates counterclockwise when viewed from the origin. (Your instructors prefer angle bracket notation < > for vectors.)

Learn how to find the vector parametric equation for a circle with radius 3, centered at (1, 4, 3) in the y = 4 plane, rotating counterclockwise when viewed from the origin. This problem covers concepts in parametric equations, vector algebra, and geometry. Ideal for advanced high school or early college students.

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