Q 13. - Calculus

Question

Explain why any two skew lines lie on a unique pair of parallel planes.

Step-by-Step Solution

Answer

Any point from line ${\mathfrak{L}}_1$ along with $\mathbf{N}$ and any point from line ${\mathfrak{L}}_2$ along with $\mathbf{N}$ determine the unique pair of parallel planes.

Step 1: Given information

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

Step 2: Calculation

Take a look at the two skew lines ${\mathfrak{L}}_1$ and ${\mathfrak{L}}_2$

The goal is to explain why two skew lines are parallel to each other on a single set of parallel lines.

Use the skew line definition to demonstrate the effect.

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

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. Assume that ${\mathbf{d}}_1$ and ${\mathbf{d}}_2$ be the direction vectors of the skew lines ${\mathfrak{L}}_1$ and ${\mathfrak{L}}_2$

Step 3: Calculation

Since, the lines are skew, therefore,

$$\begin{gathered} \mathbf{N}={\mathbf{d}}_1\times {\mathbf{d}}_2 \\ \ne 0 \end{gathered}$$

Therefore, any point from line ${\mathfrak{L}}_1$ along with $\mathbf{N}$ and any point from line ${\mathfrak{L}}_2$ along with $\mathbf{N}$ determine the unique pair of parallel planes.