Planes in Space

A plane in three-dimensional space is defined by two key components: a point on the plane and a normal vector perpendicular to its surface. The normal vector is orthogonal to the plane and uniquely determines its orientation. Together, these elements completely specify the plane's position and direction in space.

The dot product of two vectors equals zero when they are orthogonal. Since any vector lying on the plane is orthogonal to the normal vector, their dot product equals zero. This orthogonality condition forms the foundation of the vector equation of the plane: n · (r - r₀) = 0.

The vector equation of a plane is n · (r - r₀) = 0, where n is the normal vector, r₀ is a position vector of a known point on the plane, and r is any point on the plane. This equation expresses that the vector from the base point to any other point on the plane is orthogonal to the normal vector.

Expand the dot product into x-, y-, and z-components using the normal vector components and coordinates of the fixed base point. If n = ⟨a, b, c⟩ and r₀ = ⟨x₀, y₀, z₀⟩, the scalar equation becomes a(x - x₀) + b(y - y₀) + c(z - z₀) = 0, describing all points on the plane.

Engineers substitute the coordinates of proposed mounting points into the plane equation to check if they satisfy the condition. If a point's coordinates satisfy the equation, it lies on the intended plane surface. This verification ensures structural integrity and accurate installation of architectural components like glass panels.

The normal vector is perpendicular to the plane's surface and uniquely determines its orientation. Every vector lying on the plane is orthogonal to this normal vector. This orthogonality relationship is the key principle used to derive both the vector and scalar equations of the plane.

The vector difference r - r₀ represents a vector connecting a fixed base point to any other point on the plane. Since this vector lies entirely on the plane surface, it must be orthogonal to the normal vector. This relationship forms the basis for the orthogonality condition that defines the plane equation.