Coordinates and Vectors in Space Geometry

Coordinate Geometry is even more useful in space than in the plane, since it is much harder to draw figures in 3D for accurate visualization.

Notation:

Vectors will be written either as rows or columns when only vector operations are concerned. If matrices enter the picture, then there is a difference between row vectors (row matrices) and column vectors (column matrices).

• In general, one writes O or sometimes 0 for the zero vector.
• In 2-space one often writes O or 0 for (0, 0), I or i for (1, 0) and J or j for (0, 1).
• In 3-space one oftern writes O or 0 for (0, 0, 0), I or i for (1, 0, 0) and J or j for (0, 1, 0), K or k for (0, 0, 1).
• In n-space, even when n = 1, 2 or 3, one writes vectors e₁, ..., e_n, where ek is the vector in n-space with all entries = 0 except for 1 in the k entry.

Vector Addition and Scalar Multiplication

It is assumed in these notes that the mechanics and basic geometry of vector addition and multiplication of a vector by a scalar (i.e., a real number) are understood.

Centers of Mass and Parametrization of Lines and Planes

Lines and planes can be described in coordinate geometry by parametrizations. This topic was studied in Math 444. A review and a few exercises are provided at the link above.

Equations of lines and planes

Lines and planes are also the solutions sets of linear equations. This section introduces the topic.

Dot Product Basics

The basic facts about dot product are here, the definition, algebraic properties, relationship with cosine, and dot product test for orthogonal vectors.

Cross Product Basics

This site at Texas A&M defines cross product and shows some basic properties. Here is another reference to MathWorld.

Normal Vectors to Planes and Lines

The equation of a plane in space or a line in the plane has a vector form, A ^. X = k. The vector A is a normal vector to the plane (or line). The normal vector can be divided by its length to make a unit normal. The cross product can be used to find normal vectors.

Applications of Vector Methods to Spherical Geometry

All the vector tools for working with equations of planes, normal directions, as well as the angle information from dot and cross product, can be used to answer many questions on the sphere.

Applications to Computer Graphics and Computer Games

Here is one example of many: an explanation of hidden surface removal by Jeff Weeks.