Why the Line of Intersection of Two Planes is Perpendicular to the Normal Vectors of Both Planes

To understand the geometric relationship between the line of intersection of two planes and the normal vectors of these planes, let's break down the concepts involved and provide a clear explanation.

Planes and Normal Vectors

A plane in three-dimensional space can be defined by its normal vector. The normal vector is a vector that is perpendicular to the plane.

Plane 1: The plane is defined by the equation: Ax By Cz D_1 0

Normal Vector for Plane 1: mathbf{n_1} langle A, B, C rangle

Plane 2: The plane is defined by the equation: Dx Ey Fz D_2 0

Normal Vector for Plane 2: mathbf{n_2} langle D, E, F rangle

The Line of Intersection

The line of intersection of two planes is the set of points that satisfy both plane equations simultaneously. This line can be represented parametrically and lies within both planes.

Perpendicularity

The line of intersection is perpendicular to both the normal vectors of the planes. This relationship is defined by the following:

Normals and their Orientation: The normal vectors mathbf{n_1} and mathbf{n_2} provide the orientation of their respective planes.

Vector Cross Product: The direction vector of the line of intersection, denoted as mathbf{d}, can be found using the cross product of the two normal vectors: mathbf{d} mathbf{n_1} times mathbf{n_2}.

Geometric Interpretation: The line of intersection is perpendicular to both normal vectors because the normal vectors are orthogonal to their respective planes.

Mathematical Representation

Mathematically, the perpendicularity is expressed as:

mathbf{n_1} cdot mathbf{d} 0 quad text{and} quad mathbf{n_2} cdot mathbf{d} 0

This shows that the direction vector mathbf{d} of the line of intersection is perpendicular to both mathbf{n_1} and mathbf{n_2}.

Conclusion

In summary, the line of intersection of two planes is perpendicular to the normal vectors of both planes because the normal vectors define the orientation of the planes, and the line of intersection must lie in both planes making it orthogonal to the normals.

This relationship is a fundamental concept in geometry and is essential for understanding the interaction between planes in three-dimensional space.