Intersection of three planes

Intersection of Three Planes: An Overview

• The intersection of three planes can result in four possible outcomes: a single point, a line, a plane, or no intersection at all.
• To find this intersection, it involves solving equations of the planes concurrently, similar to systems of linear equations.

Intersection Process

• Given systems of linear equations representing the three planes, solve these equations simultaneously to find the intersection.
• Two planes will always intersect in a line unless they are parallel. Find this line first.
• Substitute the equation of this line into the equation of the third plane to find the point of intersection.

Possible Outcomes

• If the planes intersect at a single point, they are referred to as coincident which means all three planes meet at one point.
• If the intersect in a line, this means all three planes share a common line which is their intersection.
• If all three equations represent the same plane, the intersection is that plane itself.
• A lack of intersection suggests that at least two of the planes are parallel and do not intersect.

Understanding through Examples

• Consider the intersection of three planes represented by the following equations: x + y + z = 1, 2x - y + 3z = 4, and 4x + z = 5.
• By solving these equations concurrently, the intersection point can be determined.

Potential Challenges

• The algebra can become complicated when determining the intersection. Be sure to review simultaneous equations.
• Understanding the concept of intersection can be challenging as it involves visualising in three dimensions. Practice by sketching them can be helpful.

Mastering the intersection of three planes helps develop skills in algebra, geometry and calculus. It is crucial to the understanding of vectors and spaces making it important for the Further Pure 1 module.