## 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 e_{1},
..., 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.

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.

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

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

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

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.

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.