Lattice Geometry

Vectorial Descriptions of Planes

Just as vectors can be used to describe points, they can also be used to describe planes. By a combination of vectors and points, we can define any plane we need to. There are two methods we will use in this module, though there are several others which are equally valid.

Method 1: Point and Normal Vector

When we say "normal" in mathematics, it doesn't mean "ordinary." Instead, it means "perpendicular at a given point." So the normal vector to the floor is a line pointing straight up, while a normal vector to a sphere is a line pointing outward from its center.

We can define a plane by choosing these two things: a point which lies in the plane, and then a normal vector to that plane. Once we have chosen these two things, there is one and only one plane that fits. There are an infinite number of planes which pass through our point, but only one has the same normal vector.

Method 2: Two intersecting vectors

An alternate method is to specify one vector which lies along our plane, and then another vector which crosses it. Once we pick two intersecting lines, there is one and only one plane which contains them both. Again, an infinite number of planes contain the first vector we picked, but only one contains both of them.

The third vector - the red one - again represents the normal vector. This was added to illustrate something you'll learn about in a few pages: the cross product. When we take the cross product of the two vectors that define our plane, we get a normal vector to that plane.


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