Skew Lines and Shortest Distances: Quick Revision

A brief note on Skew Lines and Shortest Distances. Read it before a test. Try each question at the end before you open its answer.

Three kinds of pair

• Parallel: $\vec{b}_1 \times \vec{b}_2 = \vec{0}$.
• Intersecting: not parallel, and $(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0$. Solve for $\lambda$ and $\mu$ to find the point.
• Skew: not parallel, and the triple product is not $0$. Skew lines need three dimensions.

Distances

• Skew lines: $d = \dfrac{\lvert (\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) \rvert}{\lvert \vec{b}_1 \times \vec{b}_2 \rvert}$. The common perpendicular has direction $\vec{b}_1 \times \vec{b}_2$.
• Parallel lines: $d = \dfrac{\lvert (\vec{a}_2 - \vec{a}_1) \times \vec{b} \rvert}{\lvert \vec{b} \rvert}$.
• Point to a line: the foot is $F = \vec{a} + t\vec{b}$ with $t = \dfrac{(P - \vec{a}) \cdot \vec{b}}{\lvert \vec{b} \rvert^2}$, and $d = PF = \dfrac{\lvert (P - \vec{a}) \times \vec{b} \rvert}{\lvert \vec{b} \rvert}$.
• Distance from the $x$-axis: $\sqrt{y^2 + z^2}$.

A worked example

Find the shortest distance between the line through $(1, 2, 0)$ with direction $(1, -1, 1)$ and the line through $(0, 1, 2)$ with direction $(2, 1, -1)$.

1. The cross product. $(1, -1, 1) \times (2, 1, -1) = (0, 3, 3)$, with length $3\sqrt{2}$.
2. The gap. $\vec{a}_2 - \vec{a}_1 = (-1, -1, 2)$.
3. The triple product. $(-1, -1, 2) \cdot (0, 3, 3) = 3$.
4. The distance. $d = \dfrac{3}{3\sqrt{2}} = \dfrac{1}{\sqrt{2}} \approx 0.707$.

Practice with new numbers

A program to try

The program finds the shortest distance of the worked example and the foot of the perpendicular from the origin to the first line.

Mistakes to avoid

• Forgetting to divide by $\lvert \vec{b}_1 \times \vec{b}_2 \rvert$. The triple product alone is a volume.
• Using the skew formula for parallel lines. The denominator is $0$.
• Dividing by $\lvert \vec{b} \rvert$ in $t$. The parameter divides by $\lvert \vec{b} \rvert^2$.
• Trusting a flat picture. Skew lines seem to cross in a drawing.

Check yourself

1. Are the lines with directions $(1, 2, 3)$ and $(-3, -6, -9)$ parallel?

2. What is the shortest distance between the $z$-axis and the line through $(3, 4, 0)$ parallel to the $z$-axis?

3. The triple product of two lines that are not parallel is $0$. What does this say?

4. Find the foot of the perpendicular from $(2, 2, 1)$ to the line through the origin with direction $(0, 3, 4)$.

5. Why does the shortest distance between two skew lines use the cross product of the directions?

Sources

• OpenStax, Calculus Volume 3, Chapter 2, section on equations of lines and planes in space.
• NCERT, Mathematics Part II for Class XII, Chapter 11, section on the shortest distance between two lines.