Understanding the Parametric Vector Form of a Plane

In three-dimensional space, a plane can be defined using vectors - quantities that have both magnitude and direction.
The Parametric Vector Form represents each point in a plane as a linear function of two parameters.

Defining a Plane in Parametric Vector Form

A plane can be expressed in parametric vector form by r = a + λb + μc where a, b and c are vectors, λ and μ are parameters which take all real values, and r is the position vector of any point on the plane.
The vector a is a position vector locating a given point on the plane.
The vectors b and c are acting as “direction” or “parallel” vectors and are vectors in the plane itself.
Importantly, vectors b and c should not be parallel (that means, they are not scalar multiples of each other).

The parametric vector form explicitly shows all points that belong to the plane.
There are an infinite number of ways to express any given plane in parametric form, each differing by the choice of the point and the direction vectors.
The parameters λ and μ enable you to generate every point on the plane.

The problem may involve switching between different forms. Be sure to familiarize yourself with the conversion between parametric vector form and normal vector (or Cartesian) form.
When asked to find the cross product in order to reveal the normal to the plane, use b × c, assuming b and c are the direction vectors.
Remember, the knowledge of the direction cosines and angles between the planes can be useful in solving complex problems.