what do the the vector function x = 2, y = 3 cos t, z = 3 sin t for 0 ≤ t ≤ 2π describe

1. Identify the components of the vector function:
x = 2
y = 3\cos(t)
z = 3\sin(t)

2. Notice \( x = 2 \) is constant, so the motion is in the \( yz \)-plane.

3. The parametric equations \( y = 3\cos(t) \) and \( z = 3\sin(t) \) describe a circle in the \( yz \)-plane with radius 3:

3\cos(t)^2 + 3\sin(t)^2 = 9 (\cos^2(t) + \sin^2(t)) = 9

4. The point (2, 0, 0) is the fixed x-coordinate.

5. Combining these, the function represents a circle of radius 3 centered at (2, 0, 0) in the \( yz \)-plane.

Conclusion: The vector function represents a circle of radius 3 centered at the point (2, 0, 0) in the \( yz \)-plane.