Pairs of Lines in 3D

Introduction: why this matters 🎯

In three-dimensional space, lines can behave in ways that are impossible on a flat page. students, when you study pairs of lines in $3D$, you are learning how to describe whether two lines are parallel, intersecting, skew, or the same line. This is a key part of coordinate geometry and vector geometry in IB Mathematics: Analysis and Approaches HL.

By the end of this lesson, you should be able to:

• identify the possible relationships between two lines in space,
• use vectors and coordinates to test whether lines meet or are parallel,
• calculate the angle between lines using direction vectors,
• understand how line equations help solve geometry problems in $3D$,
• connect this topic to other ideas in vectors, planes, and coordinate geometry.

This lesson is important because many real-world systems use $3D$ geometry 🧭, such as navigation, architecture, robotics, and computer graphics.

1. What can happen when two lines are in $3D$?

In a plane, two lines either intersect, are parallel, or are the same line. In $3D$, there is one extra possibility: the lines can be skew. Skew lines do not intersect and are not parallel because they lie in different planes.

So the four possible relationships are:

• Intersecting lines: they meet at exactly one point.
• Parallel lines: they have the same direction and never meet.
• Coincident lines: they are actually the same line.
• Skew lines: they never meet and are not parallel.

If two lines are in the same plane, they are called coplanar. Intersecting, parallel, and coincident lines are coplanar. Skew lines are not coplanar.

Example

Suppose line $l_1$ goes through point $A(1,2,3)$ with direction vector $\mathbf{d}_1=(2,-1,4)$, and line $l_2$ goes through point $B(3,1,7)$ with direction vector $\mathbf{d}_2=(4,-2,8)$.

Notice that $\mathbf{d}_2=2\mathbf{d}_1$. This means the lines have parallel direction vectors. To decide whether they are parallel but different, or the same line, we check whether point $B$ lies on line $l_1$. If it does not, the lines are distinct parallel lines.

2. Vector form of a line in $3D$

A line in space is often written using a point and a direction vector. If a line passes through point with position vector $\mathbf{a}$ and has direction vector $\mathbf{d}$, then its vector equation is

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

where $\lambda$ is a scalar parameter.

In component form, if $\mathbf{a}=(x_0,y_0,z_0)$ and $\mathbf{d}=(a,b,c)$, then

$$x=x_0+\lambda a,\quad y=y_0+\lambda b,\quad z=z_0+\lambda c$$

This is called the parametric form of the line.

Why this is useful

The direction vector tells us the line’s direction in space, and the parameter lets us generate every point on the line. When studying pairs of lines, comparing direction vectors and solving simultaneous equations are the main tools.

Example

Consider the lines

$$l_1: \mathbf{r}=(1,0,2)+t(3,1,-1)$$

$$l_2: \mathbf{r}=(4,2,1)+s(-6,-2,2)$$

The direction vectors are $(3,1,-1)$ and $(-6,-2,2)$. Since

$$(-6,-2,2)=-2(3,1,-1)$$

the lines are parallel or coincident.

To check whether they are the same line, test whether the point $(4,2,1)$ lies on $l_1$. From $x=1+3t=4$, we get $t=1$. Then $y=0+t=1$, but the point has $y=2$, so it is not on the line. Therefore, the lines are parallel and distinct.

3. How to test whether two lines intersect ✨

To see whether two lines intersect, write both in parametric form and solve the coordinate equations simultaneously. If there is a pair of parameter values that gives the same point, then the lines intersect.

Suppose

$$l_1: \mathbf{r}=(2,-1,0)+t(1,2,3)$$

$$l_2: \mathbf{r}=(5,1,6)+s(-2,0,-3)$$

Then the coordinate equations are

$$2+t=5-2s$$

$$-1+2t=1$$

$$3t=6-3s$$

From $-1+2t=1$, we get $t=1$. Substituting into $3t=6-3s$ gives $3=6-3s$, so $s=1$. Checking the first equation: $2+1=5-2(1)$ gives $3=3$, so the lines intersect at the point $(3,1,3)$.

Important idea

Sometimes a system has no solution. Then the lines do not intersect. If their direction vectors are not parallel and there is no solution, the lines are skew.

4. Skew lines: the new $3D$ feature 😮

Skew lines are one of the biggest differences between $2D$ and $3D$. They are lines that do not intersect and are not parallel. Because they are not in the same plane, there is no single flat figure containing both.

A common mistake is to assume that two non-parallel lines must intersect. That is true in a plane, but not in space.

Example

Let

$$l_1: \mathbf{r}=(0,0,0)+t(1,0,0)$$

$$l_2: \mathbf{r}=(0,1,1)+s(0,1,0)$$

The direction vectors $(1,0,0)$ and $(0,1,0)$ are not multiples, so the lines are not parallel. To check for intersection, compare coordinates:

$$x=t$$

$$y=0$$

$$z=0$$

for $l_1$, and

$$x=0$$

$$y=1+s$$

$$z=1$$

for $l_2$.

Since $l_1$ always has $y=0$ and $z=0$, while $l_2$ always has $z=1$, there is no point in common. Therefore, the lines are skew.

5. Angle between two lines

The angle between two lines is found using their direction vectors. If the direction vectors are $\mathbf{d}_1$ and $\mathbf{d}_2$, then the angle $\theta$ between the lines is given by

$$\cos\theta=\frac{|\mathbf{d}_1\cdot\mathbf{d}_2|}{|\mathbf{d}_1||\mathbf{d}_2|}$$

The absolute value ensures the angle is the acute angle between the lines.

Example

Let $\mathbf{d}_1=(2,1,2)$ and $\mathbf{d}_2=(1,2,2)$. Then

$$\mathbf{d}_1\cdot\mathbf{d}_2=2(1)+1(2)+2(2)=8$$

and

$$|\mathbf{d}_1|=\sqrt{2^2+1^2+2^2}=3,$$

$$|\mathbf{d}_2|=\sqrt{1^2+2^2+2^2}=3$$

So

$$\cos\theta=\frac{8}{9}$$

and

$$\theta=\cos^{-1}\left(\frac{8}{9}\right)$$

This method works for both intersecting and skew lines, because the angle depends only on direction.

6. Connecting pairs of lines to the rest of geometry and trigonometry

Pairs of lines in $3D$ connect to several major ideas in the topic of Geometry and Trigonometry:

• Vectors: direction vectors describe line direction and help test parallelism.
• Coordinate geometry: parametric equations allow algebraic solutions.
• Planes: two intersecting lines determine a plane, while skew lines do not lie in one common plane.
• Angles: dot products find angles between lines and later between lines and planes.
• Problem solving: many IB questions ask for intersections, shortest distances, or verification of geometric relationships.

Real-world connection 🌍

Imagine two drones flying in space. Their paths can be modeled as lines. If the paths are parallel, the drones keep the same direction. If they intersect, there is a collision point. If they are skew, they pass by each other at different heights or depths. Mathematics helps predict these situations safely.

7. Common exam skills and mistakes

When solving IB-style questions, students, these steps are especially useful:

1. Write both lines in a clear vector or parametric form.
2. Compare direction vectors first.
3. If the vectors are scalar multiples, test whether the lines coincide.
4. If they are not parallel, solve the coordinate equations to check for intersection.
5. If there is no intersection and the lines are not parallel, conclude they are skew.

Common mistakes

• Thinking non-parallel lines must intersect in $3D$.
• Forgetting to check whether parallel lines are actually the same line.
• Mixing up points and direction vectors.
• Solving the equations incorrectly because of arithmetic errors.

Careful notation helps avoid these problems.

Conclusion

Pairs of lines in $3D$ is a core part of IB Mathematics: Analysis and Approaches HL because it shows how algebra describes geometry in space. You now know the four main relationships between two lines: intersecting, parallel, coincident, and skew. You also know how to use vector equations, parametric equations, and direction vectors to test these relationships and find the angle between lines.

This topic builds directly into more advanced geometry, especially the study of planes and shortest distances. Mastering it gives you a strong foundation for solving spatial problems accurately and efficiently ✅

Study Notes

• In $3D$, two lines can be intersecting, parallel, coincident, or skew.
• A line can be written as $\mathbf{r}=\mathbf{a}+\lambda\mathbf{d}$, where $\mathbf{a}$ is a position vector and $\mathbf{d}$ is a direction vector.
• Parallel lines have direction vectors that are scalar multiples.
• Coincident lines are the same line; a point from one line must lie on the other.
• Intersecting lines share exactly one point.
• Skew lines do not intersect and are not parallel.
• To check intersection, solve the parametric equations simultaneously.
• The angle between two lines is found using $\cos\theta=\frac{|\mathbf{d}_1\cdot\mathbf{d}_2|}{|\mathbf{d}_1||\mathbf{d}_2|}$.
• Geometry in $3D$ is different from geometry in a plane because non-parallel lines do not have to meet.
• These ideas are fundamental for later work with planes, distances, and spatial reasoning.