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 line in 3D is determined by a point and a direction. Parametric form is the most flexible representation: it naturally handles all orientations and extends to higher dimensions.

Common stuck point: Two lines in 3D can be skew—they don't intersect AND aren't parallel. This never happens in 2D. To check if lines intersect, set their parametric equations equal and solve; if there's no consistent solution, the lines are skew.

Sense of Study hint: Write the line using a known point and a direction vector. Plug in t = 0 to verify you get the known point, then try t = 1 to get a second point.

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.

Solution

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

Answer

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

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}.
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Background Knowledge

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

parametric equations