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Let's break down the problem first to ensure we handle the calculations properly.

### Part (a):
We need to determine the constant \( c \) so that the angle between the line \( g \) and the \( x_1x_2 \)-plane (i.e., the xy-plane) is 45°. The equation of the line \( g \) is given as:
\[
\vec{x} = r \cdot \begin{pmatrix} 3 \\ 4 \\ c \end{pmatrix}
\]

To find \( c \), we use the fact that the angle between the direction vector of the line \( \begin{pmatrix} 3 \\ 4 \\ c \end{pmatrix} \) and the \( x_1x_2 \)-plane (or xy-plane) can be computed using the formula for the angle between two vectors. In this case, the xy-plane has a normal vector along the \( x_3 \)-axis, i.e., \( \vec{n} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \).

The formula for the cosine of the angle \( \theta \) between the line and the plane is:
\[
\cos \theta = \frac{|\vec{d}_z|}{|\vec{d}|}
\]
where \( |\vec{d}_z| \) is the absolute value of the z-component of the direction vector \( c \), and \( |\vec{d}| \) is the magnitude of the direction vector \( \begin{pmatrix} 3 \\ 4 \\ c \end{pmatrix} \).

For \( \theta = 45^\circ \), we know:
\[
\cos 45^\circ = \frac{\sqrt{2}}{2}
\]

Using the direction vector:
\[
|\vec{d}| = \sqrt{3^2 + 4^2 + c^2} = \sqrt{9 + 16 + c^2} = \sqrt{25 + c^2}
\]
and
\[
|\vec{d}_z| = |c|
\]

Thus, we have the equation:
\[
\frac{|c|}{\sqrt{25 + c^2}} = \frac{\sqrt{2}}{2}
\]

Now, let's solve this equation for \( c \).The error occurred because we cannot directly solve equations involving the absolute value of the variable \( c \) in this form. To proceed, we can break the equation into two cases:

1. \( c \geq 0 \), so \( |c| = c \)
2. \( c < 0 \), so \( |c| = -c \)

Let's solve these two cases separately.The two possible solutions for \( c \) are:

1. \( c = 5 \)
2. \( c = -5 \)

Thus, \( c \) can either be 5 or -5 for the angle between the line and the \( x_1x_2 \)-plane to be 45°.

### Part (b):
We are asked to consider all lines through the origin that form an angle of 45° with the \( x_1x_2 \)-plane. Then, we need to describe the points of intersection of these lines with the plane \( E: x_3 = 5 \).

These lines must have direction vectors that form a 45° angle with the xy-plane, meaning their direction vector has the form \( \begin{pmatrix} a \\ b \\ c \end{pmatrix} \), where \( c = \pm\sqrt{a^2 + b^2} \). Since all lines pass through the origin, their parametric form is:
\[
\vec{x} = r \cdot \begin{pmatrix} a \\ b \\ c \end{pmatrix}
\]

Now, to find the intersection with the plane \( x_3 = 5 \), set the \( x_3 \)-coordinate of the parametric equation equal to 5:
\[
r \cdot c = 5
\]

Solving for \( r \):
\[
r = \frac{5}{c}
\]

Substituting this into the equations for \( x_1 \) and \( x_2 \):
\[
x_1 = \frac{5a}{c}, \quad x_2 = \frac{5b}{c}
\]

Therefore, the points of intersection are given by:
\[
\left( \frac{5a}{c}, \frac{5b}{c}, 5 \right)
\]

Would you like further details on these calculations, or clarification on any part? Here are 5 related questions:

1. How is the angle between a line and a plane generally calculated?
2. What would the solution be if the angle between the line and the plane was different (e.g., 30°)?
3. How do parametric equations describe lines in 3D space?
4. What role does the z-component \( c \) play in the orientation of the line relative to the plane?
5. Can the intersection points vary if the angle between the line and the plane is different?

**Tip**: When dealing with angles between lines and planes, always use the direction vector and normal vector to simplify calculations.

Bestimmen Sie c so, dass der Winkel zwischen der x1x2-Ebene und der Geraden g: x = r · (3, 4, c) die Größe 45° hat. Betrachten Sie alle Ursprungsgeraden, die mit der x1x2-Ebene einen Winkel von 45° bilden. Beschreiben Sie die Lage der Schnittpunkte dieser Geraden mit der Ebene E: x3 = 5.

Solve the problem of determining the constant 'c' such that the angle between a line and the x1x2-plane is 45°. Also, explore the parametric equations of lines forming a 45° angle with the plane and their intersection with the plane E: x3 = 5. This covers core topics in vector geometry.

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