1. **State the problem:** We need to understand and graph the parametric vector function $$r(t) = (-\sin t, -\cos t, 2t)$$ for $$t$$ in the interval $$[-\frac{3\pi}{2}, \frac{3\pi}{2}]$$. 2. **Explain the components:** The function has three components: - $$x(t) = -\sin t$$ - $$y(t) = -\cos t$$ - $$z(t) = 2t$$ 3. **Analyze the projection on the xy-plane:** The first two components describe a point on the unit circle but reflected and rotated: $$x(t)^2 + y(t)^2 = (-\sin t)^2 + (-\cos t)^2 = \sin^2 t + \cos^2 t = 1$$ This means the projection of the curve on the xy-plane is a circle of radius 1. 4. **Analyze the z-component:** The $$z$$ coordinate increases linearly with $$t$$ as $$z = 2t$$. 5. **Interpretation:** The curve is a helix wrapped around the unit circle in the xy-plane, moving upward and downward along the z-axis as $$t$$ varies. 6. **Domain:** $$t$$ ranges from $$-\frac{3\pi}{2}$$ to $$\frac{3\pi}{2}$$, so the helix makes $$\frac{3\pi}{2} - (-\frac{3\pi}{2}) = 3\pi$$ radians, or 1.5 full turns. 7. **Summary:** The parametric curve is a helix of radius 1, centered on the z-axis, with vertical stretch factor 2. **Final answer:** The graph of $$r(t) = (-\sin t, -\cos t, 2t)$$ for $$t \in [-\frac{3\pi}{2}, \frac{3\pi}{2}]$$ is a helix making 1.5 turns around the z-axis with radius 1 and vertical pitch 2.