Answer
Ellipse centered at $(0, 0, 1)$ on the plane $z = 1$
Work Step by Step
Step 1
From a vector-valued function: \[ \mathbf{r} = x(t)\mathbf{i} + y(t)\mathbf{j} + z(t)\mathbf{k} \] the parametric equations can be obtained: \[ \begin{align*} x &= x(t) \\ y &= y(t) \\ z &= z(t) \end{align*} \] The graph can be identified by remembering certain forms, using trigonometric relationships, converting to Cartesian form, or by plotting some points and drawing a smooth curve through them.
Step 2
From: \[ \mathbf{r} = 2\cos(\theta)\mathbf{i} - 3\sin(\theta)\mathbf{j} + \mathbf{k} \] we have parametric equations: \[ \begin{align*} x &= 2\cos(\theta) \\ y &= -3\sin(\theta) \\ z &= 1 \end{align*} \] With the constant $z = 1$, the action takes place on the plane $z = 1$. $x$ and $y$ are similar to the parametric form of an ellipse centered at $(0, 0)$: \[ \begin{align*} x &= a\cos(\theta) \\ y &= b\sin(\theta) \end{align*} \] $a$ and/or $b$ can be negative, which affects whether the ellipse is drawn clockwise or counterclockwise for increasing $\theta$.
Step 3
$|\mathbf{r}| = 2|a| = 2$ means the length of the ellipse axis parallel to the $x$-axis is: \[ 2|a| = 4 \] $|\mathbf{r}| = 3|b| = 3$ means the length of the ellipse axis parallel to the $y$-axis is: \[ 2|b| = 6 \] The longer one is the major axis, the other is the minor axis.
Result: Ellipse centered at $(0, 0, 1)$ on the plane $z = 1$, major axis length $6$ parallel to the $y$-axis, minor axis length $4$ parallel to the $x$-axis.
