Vector Equations of Lines and Planes Interactive Widget

A single parameter creates infinity

In three-dimensional space, you can describe an entire infinite line with just a starting point and a direction. By varying one parameter t, you trace every single point along that line. Drag the slider to move along the line and watch how the parameter dictates position.

Line Explorer t = 0.00
Drag slider to change parameter t
r = a + 0.00d
Parameter t
-3 0 +3
Position
(2, 1, 3)
Direction
d = (1, 1, 2)
Two lines are parallel if their direction vectors are scalar multiples of each other. This means one direction is just a scaled version of the other: if line 1 has direction d₁ = (2, 4, 6) and line 2 has direction d₂ = (1, 2, 3), then d₁ = 2d₂, so they're parallel. Parallel lines in 3D never meet, just like train tracks extending forever. Even though they might be at different heights or positions in space, their shared direction means they maintain constant separation. The beauty is that you can check this algebraically without needing to visualise the entire infinite lines.
Know This
A line is written as position vector = a + t d, where d is the direction vector, and different values of the parameter t trace out every point on the line.