Plans dans l'espace

A plane in three-dimensional space is defined by a point on the plane and a normal vector. This vector is orthogonal to the surface and sets the plane’s orientation.

In a real-world context, this models the surface of a sloped glass panel attached to a building.

The corner of the panel’s base lies at a known point, represented by the position vector r₀. Its orientation is set by a normal vector n orthogonal to the surface.

A second point on the panel is defined by another position vector r. Subtracting r₀ from r applies the vector difference rule and gives a vector that lies on the panel surface.

Since this vector lies on the plane, it is orthogonal to n, and their dot product equals zero.

This orthogonality forms the vector equation of the plane. Rewriting it gives another form of the same equation.

Expanding the dot product into x-, y-, and z-components gives the scalar equation. It uses the components of the normal vector and the coordinates of the fixed base point.

This equation defines all points that lie on the panel’s surface.

Modeling the panel as a plane helps check whether the mounting points satisfy the equation for structural alignment.