1. **Problem:** Given the vector function $r(t) = t\mathbf{i} + (1 - t)\mathbf{j}$ for $0 \leq t \leq 1$, describe the curve and find its key features. 2. **Formula and rules:** This is a parametric equation describing a line segment in the plane. The vector $r(t)$ gives the position at parameter $t$. When $t=0$, $r(0) = 0\mathbf{i} + 1\mathbf{j} = (0,1)$; when $t=1$, $r(1) = 1\mathbf{i} + 0\mathbf{j} = (1,0)$. 3. **Intermediate work:** The curve moves linearly from point $(0,1)$ to $(1,0)$ as $t$ goes from 0 to 1. 4. **Explanation:** This is a straight line segment connecting the points $(0,1)$ and $(1,0)$ in the xy-plane. The parameter $t$ controls the position along this line. **Final answer:** The curve is a straight line segment from $(0,1)$ to $(1,0)$ in the xy-plane.