1. The problem is to analyze the parametric vector function $$\mathbf{r}(t) = (-\sin(t), -\cos(t), 2t)$$ for $$-2 \leq t \leq 2$$. 2. This function describes a curve in 3D space where the x-coordinate is $$-\sin(t)$$, the y-coordinate is $$-\cos(t)$$, and the z-coordinate is $$2t$$. 3. The x and y components trace a circle of radius 1 in the xy-plane but with a negative sign, effectively rotating the standard circle by 180 degrees. 4. The z component increases linearly with $$t$$, so the curve is a helix wrapping around the z-axis. 5. The parameter $$t$$ ranges from $$-2$$ to $$2$$, so the curve covers a segment of the helix over this interval. 6. The function can be written as $$\mathbf{r}(t) = (-\sin(t), -\cos(t), 2t)$$, which is a standard helix with radius 1 and vertical stretch factor 2. Final answer: The curve is a helix of radius 1 centered on the z-axis, descending and ascending as $$t$$ goes from $$-2$$ to $$2$$, with vertical height $$4$$ units (since $$z=2t$$).