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.

Intersection of Line and Plane

• 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.

Intersection of Three Planes

• 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.