Equation of a Plane Given a Point and a Normal Vector

Given a point $P_0 = (1, 2, 3)$ and a normal vector $\mathbf{n} = (4, 5, 6)$, find the equation of the plane in component form.

When tasked with finding the equation of a plane given a point and a normal vector, it's essential to understand the fundamental relationship between the geometry of the plane and its algebraic representation. The position vector of any point on the plane, in combination with the normal vector, is key to constructing the plane's equation. The normal vector acts perpendicular to the plane, serving as a crucial component in delineating the plane's orientation in three-dimensional space.

The normal vector, in this context a 3D vector, allows us to define the plane in terms of all points whose position vectors create a perpendicular relationship with it. By applying the dot product between the normal vector and the vector from the given point to a variable point on the plane, we set up the necessary condition for the plane's equation. Specifically, the dot product must equal zero for the point to reside on the plane. This translates into a linear equation in three variables, describing all possible points on the plane.

Understanding this process not only involves recognizing the algebraic manipulation required to derive the equation but also appreciating the geometric interpretation of vectors and planes in 3D space. This problem emphasizes the need to master the interplay between linear algebra concepts and spatial reasoning, foundational skills in fields like physics, engineering, and computer graphics.

Related Problems