Draw the projection of the vector-valued curve
r ( t )

The plot above shows the projection of the vector-valued curve $\mathbf{r}(t) = \langle 6t^2, t, 3t^2 \rangle$ onto the $yz$-plane. The resulting curve is a parabola described by the equation $z = 3y^2$, where $y = t$ and $z = 3t^2$.

Do you have any further questions or need more details on this?

Here are 5 related questions to expand your understanding:

1. How do you project a curve onto other planes, such as the $xy$-plane or $xz$-plane?
2. What does the original 3D curve $\mathbf{r}(t)$ look like?
3. How does the projection change if the components of $\mathbf{r}(t)$ are different functions?
4. What is the geometric interpretation of vector-valued curves in 3D space?
5. Can you derive the arc length of the projected curve on the $yz$-plane?

Tip: When projecting curves onto planes, you're essentially removing one of the coordinate dependencies to analyze the curve's behavior in a reduced dimension.