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Können sie folgende Aufgabe für mich lösen?

The exercise asks you to fill in the missing values so that the point \( R \) lies on the line \( g \).

### Part a)

The line is described by the parametric equation:

\[
g: \vec{x} = 
\begin{pmatrix}
2 \\
1 \\
0
\end{pmatrix}
+ r \cdot
\begin{pmatrix}
1 \\
3 \\
-1
\end{pmatrix}
\]

The point \( R \) is \( R(-1, -3.5, \square) \), where we need to find the missing coordinate.

To find \( r \), we equate the first two coordinates of \( R \) to the corresponding coordinates in the parametric equation of the line.

1. For the first coordinate: 
\[
2 + r \cdot 1 = -1
\]
Solving for \( r \):
\[
r = \frac{-1 - 2}{1} = -3
\]

2. For the second coordinate: 
\[
1 + r \cdot 3 = -3.5
\]
Substitute \( r = -3 \):
\[
1 + (-3) \cdot 3 = -3.5
\]
\[
1 - 9 = -8
\]
This is not equal to \( -3.5 \), so there seems to be an inconsistency here. Are you sure about the value of \( -3.5 \)? Could this be a misprint?

Assuming the task is consistent, we can use the first equation to solve for the missing \( z \)-coordinate:
\[
0 + r \cdot (-1) = z
\]
Substitute \( r = -3 \):
\[
0 + (-3) \cdot (-1) = z
\]
\[
z = 3
\]

Thus, \( R \) is \( (-1, -3.5, 3) \), assuming \( -3.5 \) is correct.

---

### Part b)

The line is given by:

\[
g: \vec{x} = 
\begin{pmatrix}
3 \\
1 \\
-1
\end{pmatrix}
+ r \cdot
\begin{pmatrix}
1 \\
\text{?} \\
2
\end{pmatrix}
\]

The point \( R \) is \( R(4, 5, \text{?}) \).

1. For the first coordinate:
\[
3 + r \cdot 1 = 4
\]
Solving for \( r \):
\[
r = \frac{4 - 3}{1} = 1
\]

2. For the second coordinate, we need to find the missing value in the direction vector. Let that value be \( b \). The second coordinate gives:
\[
1 + r \cdot b = 5
\]
Substitute \( r = 1 \):
\[
1 + b = 5
\]
\[
b = 4
\]

3. For the third coordinate:
\[
-1 + r \cdot 2 = z
\]
Substitute \( r = 1 \):
\[
-1 + 2 = z
\]
\[
z = 1
\]

Thus, the direction vector is:
\[
\begin{pmatrix}
1 \\
4 \\
2
\end{pmatrix}
\]
And the point \( R \) is \( (4, 5, 1) \).

---

Do you need more details or clarification on any step?

### Related Questions:
1. How do you verify if a point lies on a line in parametric form?
2. Can you extend the parametric form to describe planes or higher-dimensional objects?
3. How would you calculate the distance between a point and a line?
4. What if a line is given in symmetric form? How do you solve it then?
5. How can you handle inconsistent systems when solving parametric equations?

### Tip:
Always double-check the consistency of given coordinates, especially when working with parametric equations, as errors can propagate through the solution.

Solve a problem to check if a given point lies on a parametric line in 3D space. Learn key concepts like parametric equations and vector algebra. Suitable for high school students in Grades 11-12.

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