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The problem you've provided is written in German and involves two parts:

**5 a)** The task asks to determine the line \( g \) that is orthogonal to the plane \( E: 4x_1 + 4x_2 - 7x_3 = 40.5 \) and passes through the point \( O(0 | 0 | 0) \), as well as to find the intersection point \( F \) of the line \( g \) with the plane \( E \).

**5 b)** The second part asks to find all points on \( g \) that have a distance of 3 units from the plane \( E \).

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Let's break down the tasks:

### Part (a): Finding the Orthogonal Line to the Plane
- The normal vector of the plane \( E \) can be directly obtained from the plane's equation. The plane equation is in the form \( ax_1 + bx_2 + cx_3 = d \), where the vector \( \vec{n} = (a, b, c) \) is the normal vector.
- The normal vector \( \vec{n} = (4, 4, -7) \) will give the direction of the line \( g \), since \( g \) is orthogonal to the plane.
- The line \( g \) passes through the origin \( O(0, 0, 0) \), so the parametric equation for \( g \) is:
  \[
  g: \vec{r}(t) = t \cdot (4, 4, -7)
  \]
  where \( t \in \mathbb{R} \).

- To find the intersection point \( F \), substitute the parametric equations of the line into the plane equation and solve for \( t \).

### Part (b): Finding Points on Line \( g \) with a Distance of 3 from the Plane
- The distance \( d \) from a point \( P(x_1, x_2, x_3) \) to a plane \( ax_1 + bx_2 + cx_3 + d = 0 \) is given by the formula:
  \[
  d = \frac{|ax_1 + bx_2 + cx_3 + d|}{\sqrt{a^2 + b^2 + c^2}}
  \]
  Using the normal vector components from part (a), you can calculate this distance for points on \( g \) and set it equal to 3 to find the specific values of \( t \).

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Would you like me to proceed with the detailed calculations for each part? If you have any specific questions, feel free to ask!

Bestimmen Sie die zur Ebene E: 4x1 + 4x2 - 7x3 = 40,5 orthogonale Gerade g durch O(0|0|0) und den Schnittpunkt F der Geraden g mit der Ebene E. Bestimmen Sie alle Punkte auf g, die von der Ebene E den Abstand 3 haben.

Learn how to find a line orthogonal to a plane and the points on the line that are a given distance from the plane. This problem involves using the normal vector of a plane, parametric equations of a line, and distance formulas. Suitable for high school students in Grades 10-12.

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