example 3 sketch the graph of each of the following vector functions.\n(a) $\\vec{r}(t) = \\langle 6 \\cos…

Explanation:

Step1: Identify parametric equations

The vector function $\vec{r}(t) = \langle 6 \cos t, 3 \sin t \rangle$ gives $x = 6 \cos t$ and $y = 3 \sin t$.

Step2: Isolate trigonometric terms

Divide by the coefficients to isolate the sine and cosine terms. $$ \cos t = \frac{x}{6}, \quad \sin t = \frac{y}{3} $$

Step3: Use Pythagorean identity

Apply the identity $\cos^2 t + \sin^2 t = 1$ to eliminate the parameter $t$. $$ \left(\frac{x}{6}\right)^2 + \left(\frac{y}{3}\right)^2 = 1 $$

Step4: Simplify the equation

Write the equation in the standard form of a conic section. $$ \frac{x^2}{36} + \frac{y^2}{9} = 1 $$

Step5: Describe the graph

The equation represents an ellipse centered at $(0,0)$ with $x$-intercepts $(\pm 6, 0)$ and $y$-intercepts $(0, \pm 3)$.

Answer:

The graph is an ellipse centered at the origin $(0,0)$ with a major axis along the $x$-axis of length 12 and a minor axis along the $y$-axis of length 6. The $x$-intercepts are $(6, 0)$ and $(-6, 0)$, and the $y$-intercepts are $(0, 3)$ and $(0, -3)$.