Planes in space can be defined using vectors as well. In fact, when defining the cross product of two vectors, we speak of the resulting vector as being perpendicular to the plane in which the original two vectors lie.

Turning this idea around, we define a plane to be the set of all vectors perpendicular to a given vector. As a result of this definition, the dot product will be of use describing the equations that yield a plane in space.

Computationally, we can quickly give an equation that characterized a plane. This is done by computing the dot product of a vector n = (a,b,c) with an arbitrary vector (x,y,z).

From step 1, we find that the equation of a plane is ax + by + cz = 0 but this is not the most general form. Planes of this type intersect the origin in space. In general, a plane may have equation

ax + by + cz = d

where d can have any value. When this occurs, the plane is being translated away from the origin by some vector r0 as depicted in the diagram below.

In general, the vector equation for a plane becomes <(r – r0),n> = 0. Here, n is the normal vector to the plane and r0 is the vector translating the plane away from the origin.

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