Question

Explain the derivation of the formula for finding the distance between two skew lines ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ Why does this formula work?

Short answer

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

Step-by-step solution

Given information

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

Calculation

The goal is to demonstrate how the formula for calculating the distance between two skew lines was developed${\mathfrak{L}}_1$ and${\mathcal{L}}_2$ respectively.

Assume the two points $P$ and $Q$ on the two skew lines ${\mathcal{L}}_1$and ${\mathcal{L}}_2$

The formula is based on computing the component vector in the given vector's direction.

The component vector is given by

$\left|{comp}_{\mathrm{N}}\overrightarrow{PQ}\right|$

The vector $\mathbf{N}$is the cross-product of the direction vectors of the two skew lines ${\mathcal{L}}_1$and ${\mathcal{L}}_2$

Calculation

The unique pair of parallel planes is determined by the skew lines. As a result of the skew in the lines,

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

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

That's why the formula works for the skew lines.