Finding the point of intersection between a line and a plane

Intersection of a Line and a Plane

Understanding the Basics

A line in three-dimensional space is typically represented by parametric equations in the form x = a + mt, y = b + nt, z = c + pt, where (a, b, c) is a point on the line and (m, n, p) is the direction of the line.
A plane can be defined by the equation ax + by + cz = d, where the plane is the collection of all points (x, y, z) that satisfy this equation.

Intersection Point

To find the point of intersection between a line and a plane, substitute the parametric equations of the line into the equation of the plane. This would yield a single equation involving the parameter t.
Solve for t. The resulting value of t represents the unique point along the line where the line intersects with the plane.
Substitute the value of t into the parametric equations of the line. This gives the x, y, and z coordinates of the intersection point.
Remember that if the line is parallel to the plane, there will be no solution for t. Consequently, no intersection point exists between the parallel line and plane.

Example

Given a line with parametric equations x = 1 + t, y = 2 - 2t, z = 3 + 3t and a plane with the equation 2x + y - z = 2, the finding the point of intersection involves:
- Substituting line equations into the plane equation to get 2(1 + t) + (2 - 2t) - (3 + 3t) = 2, which simplifies to t = -1.
- Subbing t = -1 into the line equations gives the intersection point as (0, 4, 0).