example 2 sketch the graph of each of the following vector functions.\n\n(a) $\\vec{r}(t) = \\langle t, 1…

Explanation:

Step1: Identify parametric equations

The vector function $\vec{r}(t) = \langle t, 1 \rangle$ gives the parametric equations: $$x = t, \quad y = 1$$

Step2: Eliminate the parameter

Substitute $t$ from the $x$ equation into the $y$ equation: $$y = 1$$

Step3: Determine the geometric shape

The equation $y = 1$ represents a horizontal line.

Step4: Determine the direction

As $t$ increases, $x$ increases, indicating rightward motion.

Answer:

The graph is a horizontal line passing through $(0, 1)$ with the equation $y = 1$. The orientation of the curve is from left to right as $t$ increases.