Space Coordinates

1. When three mutually perpendicular lines are drawn through a point in space, is called the origin, and the three perpendicular lines are named the -axis, -axis, and -axis, respectively. These three axes are referred to as the rectangular coordinate axes or simply coordinate axes, and the space defined by these axes is called the coordinate space.
2. The plane determined by the -axis and -axis is called the -plane, the plane determined by the -axis and -axis is called the -plane, and the plane determined by the -axis and -axis is called the -plane. These three planes are referred to as the coordinate planes.
   
   
3. A set of three real numbers corresponding to a point in coordinate space is called the space coordinates or coordinates of point , denoted as .

Coordinates of a Point in Space

1. Coordinates of the Foot of a Perpendicular
1. For a point in coordinate space, the feet of the perpendiculars from onto the -axis, -axis, and -axis are , , and , respectively.
2. The feet of the perpendiculars from onto the -plane, -plane, and -plane are , , and , respectively.
2. Coordinates of a Symmetric Point
1. The coordinates of the point obtained by reflecting across the -axis, -axis, and -axis are , , and , respectively.
2. The coordinates of the point obtained by reflecting across the -plane, -plane, and -plane are , , and , respectively.
3. The coordinates of the point obtained by reflecting across the origin are .

Distance Between Two Points

1. The distance between two points and in coordinate space is given by the formula:
   
2. The distance between the origin and a point in coordinate space is given by: