This exercise involves two lines, $g$ and $h$, in space. We are given their parametric forms and asked to:

1. Determine the intercept points of these lines with the coordinate planes and then draw them in an oblique view.
2. Prove that the lines $g$ and $h$ are skew (non-parallel and non-intersecting).

Step-by-Step Solution

Given Data

The parametric equations for the lines are:

• Line $g: \vec{x} = \begin{pmatrix} 3 \\ 3 \\ 4 \end{pmatrix} + s \cdot \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$
• Line $h: \vec{x} = \begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix} + t \cdot \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$

Part (a): Determine the Intercept Points (Spurpunkte)

To find the intercepts with the coordinate planes (where one of $x$, $y$, or $z$ is zero), we set each coordinate equal to zero in turn and solve for $s$ and $t$.

1. Intercepts of Line $g$:

   • $x = 0$: Solve $3 + s \cdot 1 = 0$.
   • $y = 0$: Solve $3 + s \cdot 1 = 0$.
   • $z = 0$: Solve $4 + s \cdot 2 = 0$.

2. Intercepts of Line $h$:

   • $x = 0$: Solve $2 + t \cdot 0 = 0$.
   • $y = 0$: Solve $2 + t \cdot 1 = 0$.
   • $z = 0$: Solve $3 + t \cdot (-1) = 0$.

After solving, plot these points to obtain a clear representation in the coordinate system.

Part (b): Prove Lines $g$ and $h$ are Skew

To show that lines $g$ and $h$ are skew, we need to check:

1. If they are parallel (they aren’t if their direction vectors are not scalar multiples of each other).
2. If they intersect by checking if a common solution exists for $s$ and $t$ when equating the parametric equations.

Parallel Check:

• Direction vector of $g$: $\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$
• Direction vector of $h$: $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$

Since these vectors are not proportional, lines $g$ and $h$ are not parallel.

Intersection Check:

Set $\begin{pmatrix} 3 + s \\ 3 + s \\ 4 + 2s \end{pmatrix} = \begin{pmatrix} 2 \\ 2 + t \\ 3 - t \end{pmatrix}$ and solve for $s$ and $t$. If no solution exists, the lines are skew.

Would you like a detailed walkthrough of each calculation step?

Additional Related Questions

1. How do you find the equation of a line passing through two points in space?
2. What conditions make two lines in space parallel, intersecting, or skew?
3. How can you find the shortest distance between two skew lines?
4. What are the geometric interpretations of intercepts in 3D space?
5. How do direction vectors affect the orientation of lines in 3D geometry?

Tip: Always start by examining direction vectors to quickly determine if lines are parallel in 3D geometry.