Question

Let ${\mathfrak{L}}_1$ and ${\mathfrak{L}}_2$ be two skew lines. Prove that no plane contains both ${\mathfrak{L}}_1$ and ${\mathfrak{L}}_2$

Short answer

There is no plane that contains two skew lines ${\mathfrak{L}}_1$and ${\mathfrak{L}}_2$

Step-by-step solution

Given information

The two skew lines ${\mathfrak{L}}_1$ and ${\mathfrak{L}}_2$

Calculation

The goal is to demonstrate that no plane contains both ${\mathfrak{L}}_1$ and ${\mathfrak{L}}_2$

Use the skew line definition to demonstrate the effect.

If the lines do not intersect or are not parallel, they are skew.

The lines ${\mathfrak{L}}_1$ and ${\mathfrak{L}}_2$ are skew. It means that lines ${\mathfrak{L}}_1$ and ${\mathfrak{L}}_2$ neither intersect nor are parallel.

Calculation

Lines in a plane either intersect or run parallel to one another.

Therefore, if the lines ${\mathfrak{L}}_1$and ${\mathfrak{L}}_2$are skew; they are not in a single plane.

Thus, there is no plane that contains two skew lines ${\mathfrak{L}}_1$and ${\mathfrak{L}}_2$