Find a parametric representation for the surface.
the plane that passes through the point (0, −1, 5) and contains the vectors 

<4, 1, 2>

 and 

<−6, 4, 5>

(Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of u and/or v.)

To find a parametric representation of a plane, we need a point on the plane and two direction vectors that lie on the plane. You are given:

- A point \( P_0 = (0, -1, 5) \)
- Two vectors that lie on the plane: 
  \[
  \mathbf{v_1} = \langle 4, 1, 2 \rangle \quad \text{and} \quad \mathbf{v_2} = \langle -6, 4, 5 \rangle
  \]

The parametric representation of the plane can be written as:

\[
\mathbf{r}(u, v) = P_0 + u \mathbf{v_1} + v \mathbf{v_2}
\]

Where \( u \) and \( v \) are parameters. Substituting the values of \( P_0 \), \( \mathbf{v_1} \), and \( \mathbf{v_2} \):

\[
\mathbf{r}(u, v) = (0, -1, 5) + u \langle 4, 1, 2 \rangle + v \langle -6, 4, 5 \rangle
\]

Now, let's break this down component-wise:

- For \( x \)-coordinate: 
  \[
  x = 0 + 4u - 6v = 4u - 6v
  \]
- For \( y \)-coordinate: 
  \[
  y = -1 + u + 4v = -1 + u + 4v
  \]
- For \( z \)-coordinate: 
  \[
  z = 5 + 2u + 5v = 5 + 2u + 5v
  \]

Thus, the parametric equations for the plane are:

\[
x = 4u - 6v, \quad y = -1 + u + 4v, \quad z = 5 + 2u + 5v
\]

Let me know if you'd like more details or have any questions!

Here are 5 related questions to expand on this topic:
1. How would you find the normal vector to this plane?
2. What is the general equation of a plane, and how does it relate to parametric equations?
3. How can we find the intersection of this plane with another surface?
4. What are the conditions for two planes to be parallel?
5. How would you find the distance between two points in 3D space?

**Tip:** In parametric representations, the two vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \) serve as the directional basis of the surface or plane.

Find a parametric representation for the surface. The plane that passes through the point (0, −1, 5) and contains the vectors <4, 1, 2> and <−6, 4, 5>. (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of u and/or v.)

Learn how to find the parametric representation of a plane passing through the point (0, −1, 5) and containing the vectors <4, 1, 2> and <−6, 4, 5>. This solution breaks down the process step by step, introducing key concepts like vector algebra and parametric equations in 3D space. Suitable for students in Grades 10-12.

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