The definition of a spherical coordinate system is a coordinate system for 3D space where the position of a point is specified by three separate numbers. These three numbers are the radial distance, polar angle, and azimuth angle.

Using the spherical coordinate system, they would be (6378 km, 40, -120).

As many as you like. Coordinate systems are arbitrary frameworks used to describe the system configuration (arrangement). The popular coordinate systems are rectangular, polar and spherical.

Coordinate systems are not the same as dimensions. A physical problem may have only one dimension and can be described in a three dimensional rectangular coordinate system. Physics since Einstein is believed to be 4 dimensional. A 4 dimensional coordinate system like quaternions would seem to be convenient.

a coordinate system is lines of longuitude and latitude that are used in a map to locate something. There is a coordiante in the Equatorial Coordinate System that is like longitude in the Geographical Coordinate System.....what is the coordinate? Celestial Equator?

I suspect the question arises from confusion. A vector itself already defines a direction, usually in the Cartesian xyz coordinate system.

If you want to express the direction in other coordinates, such as polar or spherical coordinates you need to transform the vector to these coordinate systems.

I can answer you question more fully if you can specify the specific coordinate system in which you want to know the direction.