How to Define a Plane with a Normal and a Point (Animated)

To derive the plane equation, we'll use the fact that the normal vector is perpendicular to any vector in the plane.Let's consider a general point Q with coordinates x, y, z that lies on our plane.The vector PQ from our fixed point P to any point Q on the plane must be perpendicular to the normal vector.This perpendicularity condition means their dot product must be zero.We can write this using the components of our vectors.Expanding the dot product gives us this equation.Rearranging the terms...We can simplify this to the standard form of a plane equation, where d is a constant.And d is equal to the negative dot product of the normal vector with our point P.This equation defines our plane in three-dimensional space. Every point that satisfies this equation lies on the plane.The normal vector determines the orientation of the plane, while point P fixes its position in space.Let's work through a concrete example using specific values.We'll use the point P at coordinates (1, 2, -1) and a normal vector n equal to (2, -1, 3).Starting with our general equation, we'll substitute our normal vector components.Now we'll substitute the point P to find d.This gives us our final plane equation: two x minus y plus three z plus three equals zero.Let's verify some points to see if they lie on our plane.For our first point at (0, 3, -1), let's substitute into our equation.For our second point at (-1, 1, 0), we can verify it lies exactly on the plane.If we change our normal vector, watch how the plane rotates to maintain perpendicularity.And if we change our point, the plane translates while maintaining its orientation.