What's the Difference Between Coincident, Parallel, Intersecting, and Skew Lines?

Coincident lines are the same line, parallel lines are distinct lines with the same direction, intersecting lines meet at one point, and skew lines are non-parallel lines in three dimensions that never meet.

This is IB Mathematics: Analysis and Approaches HL, sub-topic AHL 3.15: Classification of lines. It can be tested analytically on Paper 1 or using technology on Paper 2 or Paper 3.

Write two lines as

$$\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}$$

and

$$\mathbf{r}=\mathbf{c}+\mu\mathbf{d},$$

where $\mathbf{a}$ and $\mathbf{c}$ are position vectors, while $\mathbf{b}$ and $\mathbf{d}$ are direction vectors.

For example, consider

$$\mathbf{r}=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\3\end{pmatrix}, \qquad \mathbf{r}=\begin{pmatrix}2\\2\\5\end{pmatrix}+\mu\begin{pmatrix}2\\4\\6\end{pmatrix}.$$

The direction vectors are scalar multiples, and $\begin{pmatrix}2&2&5\end{pmatrix}$ lies on the first line when $\lambda =$. Therefore, the lines are coincident.

To test intersection when directions are not parallel, equate corresponding components. Solve two equations for $\lambda$ and $\mu$, then substitute into the third. If all three hold, substitute the parameter into either line to find the intersection point. If the third fails, the lines are skew.

A common misconception is that two non-parallel lines must intersect. This is true in a plane, but not in three-dimensional space: they may be skew.

Exam technique: First compare direction vectors. If they are parallel, test whether the lines share a point. If they are not parallel, solve the three component equations simultaneously: a consistent solution means intersecting; inconsistency means skew. Always state the classification and justify it mathematically.