Angle between two lines and special positions

Two lines in 3D may be:

• Intersecting , meet at a single point.
• Parallel , same direction, never meet.
• Coincident , same line.
• Skew , neither parallel nor intersecting; they live in 3D and "pass" each other.

In all cases the angle between them is determined by their direction vectors alone.

Angle formula

Let two lines have direction vectors $\vec d_1$ and $\vec d_2$. The angle $\theta$ between them satisfies $\cos\theta = \frac{|\vec d_1 \cdot \vec d_2|}{|\vec d_1|\,|\vec d_2|}.$ The absolute value forces $\theta \in [0, \pi/2]$ , the acute angle.

Equivalently, $\sin\theta = \dfrac{|\vec d_1 \times \vec d_2|}{|\vec d_1||\vec d_2|}$.

Perpendicularity

$\vec d_1 \cdot \vec d_2 = 0$. In direction-ratio form: $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$.

Parallelism

$\vec d_1$ and $\vec d_2$ are proportional: $\vec d_2 = \lambda \vec d_1$ for some $\lambda \ne 0$. Equivalently, $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$ (assuming none of the denominators are zero).

A useful equivalent test: $\vec d_1 \times \vec d_2 = \vec 0$.

Coincidence vs parallel

Two parallel lines are coincident iff they share a common point. To check: parametrise one line, plug into the other's equation, see if a consistent solution exists.

Skewness

Two non-parallel lines in 3D may or may not intersect. They intersect iff they have a common point; otherwise they are skew. The most reliable test: do the lines' direction vectors and the displacement $\vec a_2 - \vec a_1$ between their reference points lie in a common plane?

That is, the lines intersect iff $[\vec a_2 - \vec a_1\;\;\vec d_1\;\;\vec d_2] = 0.$ Non-zero triple product $\Rightarrow$ the lines are skew (assuming they are not parallel).

Worked examples

Example 1. Find the angle between the lines $\dfrac{x - 1}{2} = \dfrac{y + 1}{2} = \dfrac{z - 2}{-1}$ and $\dfrac{x - 1}{1} = \dfrac{y + 1}{-2} = \dfrac{z}{2}$.

Direction ratios: $(2, 2, -1)$ and $(1, -2, 2)$. $\vec d_1\cdot\vec d_2 = 2 - 4 - 2 = -4$. Magnitudes: $\sqrt{4 + 4 + 1} = 3$ and $\sqrt{1 + 4 + 4} = 3$.

$\cos\theta = \dfrac{|-4|}{3 \cdot 3} = \dfrac{4}{9}$.

Example 2. Are the lines $\dfrac{x - 1}{1} = \dfrac{y}{2} = \dfrac{z + 1}{3}$ and $\dfrac{x + 2}{1} = \dfrac{y - 2}{2} = \dfrac{z + 4}{3}$ parallel?

Both have direction $(1, 2, 3)$. Yes, parallel.

Example 3. Show that the lines $\dfrac{x - 1}{2} = \dfrac{y - 2}{3} = \dfrac{z - 3}{4}$ and $\dfrac{x - 4}{5} = \dfrac{y - 1}{2} = \dfrac{z}{1}$ are not parallel. Find whether they intersect.

Directions: $(2, 3, 4)$ and $(5, 2, 1)$. Not proportional (e.g. $2/5 \ne 3/2$), so not parallel.

Test intersection: $\vec a_2 - \vec a_1 = (4 - 1, 1 - 2, 0 - 3) = (3, -1, -3)$. Triple product: $\det \begin{pmatrix} 3 & -1 & -3 \\ 2 & 3 & 4 \\ 5 & 2 & 1 \end{pmatrix} = 3(3 - 8) + 1(2 - 20) + (-3)(4 - 15) = -15 - 18 + 33 = 0$.

Triple product zero, so they intersect.

Example 4. Are the lines $\dfrac{x - 1}{1} = \dfrac{y - 2}{1} = \dfrac{z - 3}{1}$ and $\dfrac{x}{2} = \dfrac{y}{2} = \dfrac{z - 1}{1}$ skew?

Directions: $(1, 1, 1)$ and $(2, 2, 1)$. Not parallel. $\vec a_2 - \vec a_1 = (-1, -2, -2)$. Triple product: $\det \begin{pmatrix} -1 & -2 & -2 \\ 1 & 1 & 1 \\ 2 & 2 & 1 \end{pmatrix}$. Expand along first row: $-1(1 - 2) - (-2)(1 - 2) + (-2)(2 - 2) = 1 - 2 + 0 = -1 \ne 0$. Skew.

Example 5. Find $\lambda$ if the lines $\dfrac{x - 2}{3} = \dfrac{y + 1}{1} = \dfrac{z + 5}{\lambda}$ and $\dfrac{x}{1} = \dfrac{y + 2}{-2} = \dfrac{z - 1}{1}$ are perpendicular.

Dot products of directions $(3, 1, \lambda)$ and $(1, -2, 1)$: $3 - 2 + \lambda = 1 + \lambda = 0$, so $\lambda = -1$.

Example 6. Find $\theta$ between $\dfrac{x}{1} = \dfrac{y}{2} = \dfrac{z}{2}$ and $\dfrac{x}{2} = \dfrac{y}{2} = \dfrac{z}{1}$.

Dot product: $2 + 4 + 2 = 8$. Magnitudes: $\sqrt 9 = 3$ each. $\cos\theta = 8/9$.

Try it yourself

1. Angle between lines with direction ratios $(1, 1, 1)$ and $(1, -1, 1)$.
2. Find $\lambda$ if $(2, 1, \lambda)$ and $(1, -2, 3)$ are perpendicular directions.
3. Are the lines $\dfrac{x}{1} = \dfrac{y}{1} = \dfrac{z}{1}$ and $\dfrac{x - 1}{1} = \dfrac{y - 1}{1} = \dfrac{z - 1}{1}$ coincident?
4. Show that the $x$- and $y$-axes are perpendicular.
5. Test whether the lines $\vec r = (1, 0, 0) + t(1, 1, 0)$ and $\vec r = (0, 0, 1) + s(0, 1, 1)$ intersect or are skew.
6. Are lines $(1, 2, 3) + t(2, 3, 4)$ and $(0, 0, 0) + s(4, 6, 8)$ parallel?
7. Find $\sin\theta$ between lines with direction ratios $(1, 0, 0)$ and $(1, 1, 0)$.
8. Find the equation of a line through $(1, 2, 3)$ perpendicular to both $\dfrac{x}{1} = \dfrac{y}{2} = \dfrac{z}{3}$ and $\dfrac{x}{2} = \dfrac{y}{1} = \dfrac{z}{-1}$.
9. Are $\vec r = (1, 1, 1) + t(2, 0, -1)$ and $\vec r = (0, 0, 0) + s(0, 1, 0)$ skew?
10. The line $L_1$ has direction $(a, b, c)$; $L_2$ has direction $(c, a, b)$. Find the angle.
11. Show that if $L_1 \parallel L_2$ and $L_1 \perp L_3$, then $L_2 \perp L_3$.
12. Are the lines $\dfrac{x - 1}{2} = \dfrac{y - 2}{3} = \dfrac{z - 3}{4}$ and $\dfrac{x - 2}{4} = \dfrac{y - 4}{6} = \dfrac{z - 6}{8}$ parallel or coincident?
13. Find the point of intersection (if any) of $\vec r = (1, 2, 3) + t(1, 1, 1)$ and $\vec r = (4, 5, 6) + s(2, -1, 1)$.
14. The body diagonals of a cube , find the angle between any two.

Pitfalls and tricks

• Always use absolute value in the angle formula , angles between lines are taken in $[0, \pi/2]$.
• Parallel lines have a single point distinguishing them (or they're coincident). Test by trying to satisfy both parametric equations simultaneously.
• Skew lines don't intersect and aren't parallel , peculiar to 3D and beyond.
• Triple product zero $\Leftrightarrow$ the two lines are coplanar (either intersecting or parallel).
• For coincidence check that a point of one is on the other.