Draw the projection of the vector-valued curve  
r
(
t
)
=
⟨
6
t
2
,
t
,
3
t
2
⟩
  on the  
y
z
 -plane.

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.

Draw the projection of the vector-valued curve r(t) = ⟨6t^2, t, 3t^2⟩ on the yz-plane.

Explore how to project the vector-valued curve r(t) = ⟨6t^2, t, 3t^2⟩ onto the yz-plane, resulting in a parabola described by z = 3y^2. This problem highlights the concepts of vector curves, parametric equations, and plane projections. Suitable for advanced high school or college-level calculus students.

Math Problem Statement

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

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