Lines in 3D Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Lines in 3D.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Lines in three-dimensional space described using parametric equations $x = x_0 + at$ , $y = y_0 + bt$ , $z = z_0 + ct$ , or symmetric form $\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$ , where $(x_0, y_0, z_0)$ is a point on the line and $\langle a, b, c \rangle$ is the direction vector.

In 2D, a line is defined by a slope and a point ( $y = mx + b$ ). In 3D, slope doesn't work—there's no single number for direction in space. Instead, you specify a starting point and a direction vector (an arrow pointing along the line). The parameter $t$ acts like a slider: at $t = 0$ you're at the starting point, and as $t$ increases or decreases, you slide along the line in the direction of the vector.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A 3D line is a starting point and a direction vector traced out by a slider parameter $t$ .

Common stuck point: The procedure for lines in 3d is the easy part; the trap is confusing the point with the direction vector. Asking "Does the line live in $x,y,z$ space and need a direction vector rather than a single slope?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the line live in $x,y,z$ space and need a direction vector rather than a single slope?

Worked Examples

Example 1

easy

Write parametric equations for the line through the point

(2, -1, 3)

in the direction of the vector

\langle 4, 1, -2 \rangle

Answer

x = 2 + 4t, \quad y = -1 + t, \quad z = 3 - 2t

First step

The parametric equations for a line through

(x_0, y_0, z_0)

with direction vector

\langle a, b, c \rangle

are:

x = x_0 + at

y = y_0 + bt

z = z_0 + ct

Full solution

2
Substitute: $x = 2 + 4t$ , $y = -1 + t$ , $z = 3 - 2t$ .
3
Verify: at $t = 0$ , the point is $(2, -1, 3)$ ✓, and the direction is $\langle 4, 1, -2 \rangle$ ✓.

A line in 3D is determined by a point and a direction vector. Parametric form represents each coordinate as a linear function of the parameter

t

. As

t

varies over all real numbers, the equations trace the entire line.

Example 2

medium

Find parametric equations for the line through

(1, 0, -2)

and

(3, 4, 1)

Example 3

medium

Convert the parametric line

x = 1 - 2t,\ y = 3 + t,\ z = 5 + 4t

to symmetric form.

Example 4

hard

Determine whether

L_1: (t, 1+t, 2-t)

and

L_2: (1+2s, 3, 1+s)

are parallel, intersecting, or skew.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Determine whether the lines

\ell_1: x = 1+t, y = 2-t, z = 3+2t

and

\ell_2: x = 3+2s, y = -s, z = 7+4s

are parallel, intersecting, or skew.

Example 2

hard

Find the distance between the parallel lines

\ell_1: \frac{x-1}{2} = \frac{y}{1} = \frac{z+1}{-1}

and

\ell_2: \frac{x-3}{2} = \frac{y-1}{1} = \frac{z}{-1}

Example 3

easy

Write parametric equations for the line through

(1,2,3)

with direction

\langle 4,5,6 \rangle

Example 4

easy

Find a point on the line

x = 2+t,\ y = -1+2t,\ z = 3t

t = 1

Example 5

easy

What is the direction vector of

x = 1-2t,\ y = 4+t,\ z = 5

Example 6

easy

Write the symmetric form of the line through

(0,1,2)

with direction

\langle 3,4,5 \rangle

Example 7

easy

Are the lines with directions

\langle 2,4,6 \rangle

and

\langle 1,2,3 \rangle

parallel?

Example 8

easy

A line has direction

\langle 0, 1, 0 \rangle

through

(2,0,5)

. Why can't you write full symmetric form?

Example 9

easy

Find where the line

x=t,\ y=2t,\ z=3t

meets the

xy

-plane.

Example 10

easy

Is the point

(5,3,1)

on the line

x=1+2t,\ y=1+t,\ z=1

Example 11

medium

Find parametric equations of the line through

(1,0,2)

and

(3,4,2)

Example 12

medium

Do the lines

L_1: (t, 2t, 3t)

and

L_2: (1+s, 2+s, 3+s)

intersect?

Example 13

medium

Show that

L_1:(t, 0, 0)

and

L_2:(0, s, 1)

are skew.

Example 14

medium

Find the point where

x=1+t,\ y=2-t,\ z=t

meets the plane

z=4

Example 15

medium

Convert symmetric form

\frac{x-1}{2} = \frac{y+3}{-1} = \frac{z}{4}

to parametric form.

Example 16

medium

Find a direction vector for the line of intersection of the planes' normals

\langle1,0,0\rangle

and

\langle0,1,0\rangle

via cross product.

Example 17

medium

Does the point

(2,2,2)

lie on

x=t,\ y=2t,\ z=3t

Example 18

medium

Find parametric equations for the line through

(0,0,0)

parallel to

\langle1,-1,2\rangle

Example 19

medium

Find the midpoint of the segment of the line

x=t,\ y=2t,\ z=t

between

t=0

and

t=2

Example 20

challenge

Find the point of intersection of

L_1:(1+t, 2t, 3-t)

and the plane

2x + y - z = 5

Example 21

challenge

Determine whether

L_1:(t,1+t,2+t)

and

L_2:(2-s, 3-s, 4-s)

intersect, are parallel, or are skew.

Example 22

challenge

Find the distance from the point

(0,0,0)

to the line

x=1+t,\ y=1,\ z=1

(direction

\langle1,0,0\rangle

Example 23

easy

Write parametric equations for the line through

(0, 0, 0)

with direction

\langle 1, 2, 3 \rangle

Example 24

easy

Find the point on the line

x = 1 + 2t,\ y = 3 - t,\ z = 4t

when

t = 2

Example 25

easy

Is the point

(7, 0, 5)

on the line

x = 3 + 2t,\ y = 4 - 2t,\ z = 1 + 2t

Example 26

easy

Write symmetric equations of the line through

(2, 1, -3)

with direction

\langle 1, 4, 2 \rangle

Example 27

easy

Find a direction vector for the line through

(1, 1, 1)

and

(4, 5, 7)

Example 28

medium

Find parametric equations for the line through

(2, -1, 4)

and

(5, 3, -2)

Example 29

medium

Find where the line

x = 2 + t,\ y = -1 + 2t,\ z = 3 - t

crosses the

xy

-plane.

Example 30

medium

Determine whether the lines

L_1: (1+t, 2-t, 3+2t)

and

L_2: (2+s, 1+s, 5)

are parallel, intersecting, or skew.

Example 31

medium

Find where the line

x = 1 + 2t,\ y = -t,\ z = 3 + t

meets the

xz

-plane.

Example 32

medium

A line has direction

\langle 2, 0, -3 \rangle

and passes through

(1, 4, 5)

. Write parametric equations.

Example 33

medium

Find the value of

t

at which the line

x = 3t,\ y = 1 + t,\ z = 2 - t

has

z = 0

. What point is that?

Example 34

medium

Write parametric equations of the line through

(0, 2, -1)

parallel to the line

x = 1 + 4t,\ y = 3 - t,\ z = 5t

Example 35

hard

Find the intersection of the line

x = 1 + t,\ y = 2t,\ z = 3 + t

with the plane

x + y - z = 4

Example 36

hard

Show that the lines

L_1: (2+t, 1-t, 3+t)

and

L_2: (1+2s, 4-2s, 2+2s)

are parallel and find a vector connecting them.

Example 37

hard

Find parametric equations for the line through

(1, 2, 3)

perpendicular to both

\langle 1, 0, 0 \rangle

and

\langle 0, 1, 0 \rangle

Example 38

hard

Find the distance from the point

(1, 2, 3)

to the line

x = t,\ y = t,\ z = t

Example 39

hard

Find parametric equations of the line through

(2, 3, 5)

perpendicular to the plane

2x - y + 3z = 1

Example 40

hard

Lines

L_1: (t, 0, 0)

and

L_2: (s, 1, 1)

have the same direction

\langle 1, 0, 0 \rangle

. Find the distance between them.

Example 41

hard

Find the point on the line

x = 2t,\ y = 1 + t,\ z = -t

closest to the origin.

Example 42

challenge

Show that the line

L: (1+t, 2-t, 3+t)

is parallel to the plane

x + 2y + z = 10

, and find the distance from the line to the plane.

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Related Concepts

Parametric Equations Planes in 3D

Background Knowledge

These ideas may be useful before you work through the harder examples.

parametric equations