2.5 Equations of lines and planes in space  (Page 4/19)

Equations for a plane

We know that a line is determined by two points. In other words, for any two distinct points, there is exactly one line that passes through those points, whether in two dimensions or three. Similarly, given any three points that do not all lie on the same line, there is a unique plane that passes through these points. Just as a line is determined by two points, a plane is determined by three.

This may be the simplest way to characterize a plane, but we can use other descriptions as well. For example, given two distinct, intersecting lines, there is exactly one plane containing both lines. A plane is also determined by a line and any point that does not lie on the line. These characterizations arise naturally from the idea that a plane is determined by three points. Perhaps the most surprising characterization of a plane is actually the most useful.

Imagine a pair of orthogonal vectors that share an initial point. Visualize grabbing one of the vectors and twisting it. As you twist, the other vector spins around and sweeps out a plane. Here, we describe that concept mathematically. Let $\text{n}=⟨a,b,c⟩$ be a vector and $P=\left(x_0,y_0,z_0\right)$ be a point. Then the set of all points $Q=\left(x,y,z\right)$ such that $\overrightarrow{PQ}$ is orthogonal to $\text{n}$ forms a plane ( [link] ). We say that $\text{n}$ is a normal vector    , or perpendicular to the plane. Remember, the dot product of orthogonal vectors is zero. This fact generates the vector equation of a plane    : $\text{n}·\overrightarrow{PQ}=0.$ Rewriting this equation provides additional ways to describe the plane:

As described earlier in this section, any three points that do not all lie on the same line determine a plane. Given three such points, we can find an equation for the plane containing these points.