example 6 sketch the graph of the following vector function.\n$$\\vec{r}(t) = \\langle 4 \\cos t, 4 \\sin t…

Explanation:

Step1: Identify parametric equations

Extract the individual components for $x$, $y$, and $z$ from the vector function $\vec{r}(t)$. $$x = 4 \cos t, \quad y = 4 \sin t, \quad z = t$$

Step2: Determine the projection on the xy-plane

Square and add the $x$ and $y$ components to find the relationship between them. $$x^2 + y^2 = (4 \cos t)^2 + (4 \sin t)^2 = 16(\cos^2 t + \sin^2 t) = 16$$

Step3: Analyze the vertical motion

Observe how the $z$ component changes as the parameter $t$ increases. $$z = t$$

Step4: Describe the 3D shape

The projection is a circle of radius 4, while the height increases linearly with $t$. The graph is a right circular helix winding around the $z$-axis.

Answer:

The graph of the vector function $\vec{r}(t) = \langle 4 \cos t, 4 \sin t, t \rangle$ is a right circular helix. It lies on the surface of the cylinder $x^2 + y^2 = 16$, has a radius of 4, and rises along the $z$-axis as $t$ increases.