Equation of a Straight Line in Space: Concepts and Examples

In 3D, a line is fixed by one point it passes through and its direction. If the point A has position vector a and the direction vector is b, then every point on the line has position vector r = a + λb, where λ is a real parameter.

This vector equation expands into parametric and symmetric (Cartesian) forms, which are the practical tools for finding points on the line, checking whether a point lies on it, and studying the line's relation to other lines and planes.

Key Concepts

• Vector form: r = a + λb, with λ ranging over all real numbers. The parameter λ slides the point along the line; λ = 0 gives the point A.
• Direction vector b = (a, b, c) fixes the line's direction; any non-zero scalar multiple of b describes the same line.
• Parametric form: x = x1 + λa, y = y1 + λb, z = z1 + λc, where (x1, y1, z1) is the known point.
• Symmetric (Cartesian) form: (x − x1)/a = (y − y1)/b = (z − z1)/c, obtained by eliminating λ. If a direction component is zero, say a = 0, write x = x1 alongside the remaining equal ratios.
• A line through two points P(x1, y1, z1) and Q(x2, y2, z2) uses the direction vector (x2 − x1, y2 − y1, z2 − z1).

Formulas

• Vector form: r = a + λb.
• Parametric form: x = x1 + λa, y = y1 + λb, z = z1 + λc.
• Symmetric form: (x − x1)/a = (y − y1)/b = (z − z1)/c.
• Two-point form: (x − x1)/(x2 − x1) = (y − y1)/(y2 − y1) = (z − z1)/(z2 − z1).

Worked Example

Write the equations of the line through (2, −1, 3) with direction (1, 2, −2), and find the point at λ = 3.

1. Vector form: r = (2, −1, 3) + λ(1, 2, −2).
2. Parametric: x = 2 + λ, y = −1 + 2λ, z = 3 − 2λ.
3. Symmetric: (x − 2)/1 = (y + 1)/2 = (z − 3)/(−2).
4. Point at λ = 3: (2 + 3, −1 + 6, 3 − 6) = (5, 5, −3).
5. Check it lies on the line: (5 − 2)/1 = 3, (5 + 1)/2 = 3, (−3 − 3)/(−2) = 3. All three ratios equal 3. Correct.

Common Misconceptions

• A line in space needs two points. One point plus a direction vector is enough; the direction can come from anywhere, not only from a second point.
• The symmetric form always works as written. If a direction component is zero, that ratio is meaningless — instead set that coordinate equal to the point's coordinate.
• Different direction vectors give different lines. Any non-zero multiple of b gives the same line, so (2, 4, −4) describes the same line as (1, 2, −2).

Real-World Applications

• Ray tracing in computer graphics: light rays are lines in parametric form.
• Flight and robot-arm path planning in 3D space.
• Finding intersections of lines with planes and with other lines.
• Describing edges of 3D models in CAD software.

A line in space is r = a + λb — one point, one direction, one parameter. From it flow the parametric and symmetric forms; remember to handle zero direction components, and that scaling the direction vector changes nothing.