Intersections

Intersection of Two Lines

Two lines are said to intersect when they cross each other at a single point, also known as the point of intersection.
In a 3-dimensional space, two lines will intersect if they are not parallel or skew.
In vector terms, two lines r1 = a + λb and r2 = c + μd intersect if there is a common solution for λ and μ that makes r1 = r2.
To find the point of intersection, equate the vector equations of line and solve for λ and μ. Thus the common point exists if a unique solution can be found.

A line and a plane intersect when there is a common point that lies both on the line and in the plane.
The intersection of a line and a plane is essentially a simultaneous solution of the vector form equations representing the line and the plane.
To find the point of intersection, substitute the vector equation of the line into the equation of the plane. If a solution exists, this gives the point on the line that also sits in the plane.
If no solution can be found, the line and the plane do not intersect and are therefore parallel.

Three planes can intersect in several ways: at a single point, along a line, along an intersection plane, or there may be no intersection if the planes are parallel or coincident.
A unique intersection point implies the planes have different normal vectors and therefore are not parallel to each other.
To find the intersection point, equate the vector form equations of the planes and solve for x, y, and z. This simultaneous solution will give the intersection point, if it exists.
If the equations do not have a unique solution, it could mean that the planes are parallel, coincident, or intersect in a line or plane. Additional analysis is required to determine which.
The algebra can be easier if the equations can first be written in the Cartesian form ax + by + cz = d.