example 5 sketch the graph of the following vector function.\n\n$$\\vec{r}(t) = \\langle 2 \\cos t, 2 \\sin…

Explanation:

Step1: Identify parametric equations

The vector function $\vec{r}(t) = \langle 2 \cos t, 2 \sin t, 3 \rangle$ gives: $$x = 2 \cos t, \quad y = 2 \sin t, \quad z = 3$$

Step2: Determine the relationship between $x$ and $y$

Square and add the $x$ and $y$ components: $$x^2 + y^2 = (2 \cos t)^2 + (2 \sin t)^2 = 4(\cos^2 t + \sin^2 t) = 4$$

Step3: Analyze the $z$ component

The $z$ coordinate is constant: $$z = 3$$

Step4: Describe the geometric shape

The equation $x^2 + y^2 = 4$ represents a circle with radius $2$ centered on the $z$-axis. Since $z = 3$, the graph is a circle in the horizontal plane $z = 3$.

Answer:

The graph is a circle of radius $2$ centered at $(0, 0, 3)$ lying in the horizontal plane $z = 3$. To sketch it, draw a circle with radius $2$ parallel to the $xy$-plane, shifted $3$ units upward along the $z$-axis.