https://r2.erweima.ai/mathbot/image/1726155678312d4inneab.jpg

Let's go through the tasks shown in the image you uploaded.

### Task (a)
You need to determine the equation of a plane, given:
- The normal vector \( \vec{n} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix} \),
- The plane passes through the point \( P(1|2|9) \).

The general form of the equation of a plane is:
\[
n_1 (x - x_0) + n_2 (y - y_0) + n_3 (z - z_0) = 0
\]
where \( (x_0, y_0, z_0) \) is a point on the plane, and \( \vec{n} = (n_1, n_2, n_3) \) is the normal vector.

Using the given information:
\[
3(x - 1) - 2(y - 2) + 1(z - 9) = 0
\]

Expanding this:
\[
3x - 3 - 2y + 4 + z - 9 = 0
\]
Simplifying:
\[
3x - 2y + z - 8 = 0
\]

Thus, the equation of the plane is:
\[
3x - 2y + z = 8
\]

### Task (b)
You are given the parametric form of the plane \( E \):
\[
E: \vec{x} = \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} + r \cdot \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + s \cdot \begin{pmatrix} -1 \\ 0 \\ 5 \end{pmatrix}
\]

To convert this to the Cartesian form, we proceed as follows:

1. Write out the parametric equations:
   \[
   \vec{x} = \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} + r \cdot \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + s \cdot \begin{pmatrix} -1 \\ 0 \\ 5 \end{pmatrix}
   \]
   This leads to the following system of equations for \( x \), \( y \), and \( z \):
   \[
   x = 1 - s
   \]
   \[
   y = -1 + r
   \]
   \[
   z = 5s
   \]

2. Express \( r \) and \( s \) in terms of \( x \), \( y \), and \( z \):
   \[
   s = 1 - x
   \]
   \[
   r = y + 1
   \]
   From \( z = 5s \), we can write:
   \[
   s = \frac{z}{5}
   \]

3. Substituting \( s = \frac{z}{5} \) into \( s = 1 - x \):
   \[
   \frac{z}{5} = 1 - x
   \]
   Multiply by 5:
   \[
   z = 5(1 - x)
   \]
   Simplifying:
   \[
   z = 5 - 5x
   \]

Thus, the Cartesian form of the plane is:
\[
z = 5 - 5x
\]

### Summary:
- The equation of the first plane is \( 3x - 2y + z = 8 \).
- The Cartesian equation of the second plane is \( z = 5 - 5x \).

Would you like further clarification or details on any step?

Here are 5 related questions to deepen your understanding:
1. How do you find the normal vector from a plane's equation?
2. How does the parametric equation of a plane relate to its Cartesian form?
3. What does the vector equation of a plane represent geometrically?
4. Can you derive a plane's equation from three points in space?
5. How do you find the intersection of two planes?

**Tip:** When converting parametric equations to Cartesian form, always look to eliminate parameters by solving for one variable in terms of another.

Geben Sie eine Koordinatengleichung für die Ebene an, die \( \vec{n} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix} \) als einen Normalenvektor besitzt und den Punkt \( (1|2|9) \) enthält. Geben Sie für die Ebene eine Koordinatengleichung an.

Learn how to find the equation of a plane using a normal vector \( \vec{n} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix} \) and a point (1|2|9). This problem covers key geometry concepts, including the formula for a plane in 3D space. Suitable for high school and early university students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation