Lines in Three Dimensions

A line in three-dimensional space is determined by one point and one nonzero direction vector.

Three coordinates describe independent movement along three perpendicular axes. A point in space is connected to the coordinate planes by dashed projections. Component arrows build its position one axis at a time.

A spatial line is every point reached by starting at one known point and moving any scalar amount along a fixed direction.

Vector equation

If a line passes through point $P_0(x_0,y_0,z_0)$ with nonzero direction vector

\vec d=\langle a,b,c\rangle,

then its vector equation is

\vec r=\vec r_0+t\vec d,qquad t\in\mathbb R.

The parameter $t$ moves forward or backward along the line. Different values produce different points.

Symmetric form

When $a,b,c$ are all nonzero, eliminate $t$:

\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}.

If a direction component is zero, use a fixed coordinate instead. For example, $a=0$ means $x=x_0$ along the entire line, so division by $a$ is not allowed.

Worked example: write all forms

At $t=0$, the line contains the required point.

Relationships between two lines

In three dimensions, two lines may:

intersect at one point;
be parallel and distinct;
coincide;
be skew, meaning nonparallel and nonintersecting.

Skew lines are possible because they can pass at different levels without sharing a plane.

Common mistakes

Using the position point as the direction. A line needs both, and they serve different roles.

Assigning a different parameter to each coordinate. One line uses one shared parameter.

Dividing by a zero direction component in symmetric form. Write that coordinate as constant.

Calling all nonintersecting lines parallel. In space they may be skew.

Stopping after two coordinate equations agree. Verify the third.

Quick self-check

Is the direction vector nonzero?
Does $t=0$ recover the stated point?
Do vector, parametric, and symmetric forms describe the same line?
Does a tested point use one consistent parameter?
For two lines, were parallel, coincident, intersecting, and skew cases distinguished?
Is the angle or distance interpretation based on direction vectors?