A line in 3D space is defined by a point and a direction vector. Any point on the line thus satisfies the parametric equations:

Parametric Equations

From the vector equation, we can derive the parametric equations for the line:

Symmetric Equations

If all , , and are non-zero, we can express the line in symmetric form:

Usages

Intersection of Two Planes

The intersection of two non-parallel planes in 3D space can be found by solving their equations simultaneously, resulting in a line.

Example

Find the intersection of the planes given by the equations

Let (parameter):

From , . Substitute into :

So the intersection line is:

Normal Vectors to a Line

The direction vector of a line determines its orientation. Any vector normal to the line satisfies the equation

Examples

Reverse Synthesis

Suppose a line is given by the system of equations

What was the vector and point that were used to define this line

Intersection Between Two Lines

Consider the two lines

Find the point of intersection of the two lines

Solving for with the third equation

Now plugging this into the second equation

Using this value with our first line we can determine the following coordinates

Intersection with Planes

Consider the line which passes through the point , and which is parallel to the line defined by , , and . Find the point of intersection of this new line with each of the coordinate planes , , and .

Determining the line we get

Now finding the intersections with the planes

Intersection with Planes

Find the point where the line defined by , , and intersects the plane