Relationships Between Two Lines
Relationships Between Two Lines
For \(l_1:\mathbf r=\mathbf a_1+\lambda\mathbf d_1\) and \(l_2:\mathbf r=\mathbf a_2+\mu\mathbf d_2\), classify directions first, then test consistency of the position equations.
Parallel lines
Skew lines
Intersecting lines
Acute angle between lines
Distance between parallel lines
Intersecting lines in a plane
Skew-line distance
| Direction test | Position test | Classification |
|---|---|---|
| \(\mathbf d_1\parallel\mathbf d_2\) | \(\mathbf a_2-\mathbf a_1\parallel\mathbf d_1\) | Coincident |
| \(\mathbf d_1\parallel\mathbf d_2\) | Connector not parallel | Distinct parallel |
| Directions not parallel | Three component equations consistent | Intersecting |
| Directions not parallel | Equations inconsistent | Skew |
\[\cos\theta=\frac{|\mathbf d_1\cdot\mathbf d_2|}{|\mathbf d_1||\mathbf d_2|},\quad 0\leq\theta\leq\frac\pi2;\qquad D_{\parallel}=\frac{|(\mathbf a_2-\mathbf a_1)\times\mathbf d|}{|\mathbf d|};\qquad D_{\rm skew}=\frac{|(\mathbf a_2-\mathbf a_1)\cdot(\mathbf d_1\times\mathbf d_2)|}{|\mathbf d_1\times\mathbf d_2|}.\]
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