Spatial right angle coordinate system

synonym space coordinates Generally refer to the spatial right angle coordinate system

definition

space arbitrarily selected a little o, Ok, OX, OY, OZ, each with each other Original point and has the same length unit. These three axes are referred to as an X-axis (horizontal axis), Y-axis (longitudinal axis), Z-axis (vertical axis), collectively referred to as a coordinate axis. Their positive direction meets the right hand rules, that is, holding the Z axis with the right hand, when the forward direction of the right hand is forwarded in the forward direction of the

angle, the point of thumbs is the Z-axis Positive direction. This constitutes a spatial right angle coordinate system, called spatial right angle coordinate system O-XYZ. The home point O is called the origin of the coordinate system. Corresponding to it is the right angle coordinate system of the left hand space. Generally, in mathematics, the right-handed coordinate system is often used in the right hand space, and the application is conveniently varied in other disciplines.

Any two coordinate axes determine a plane so that three mutually perpendicular planes are determined, which is collectively called a coordinate surface. The coordinate surface determined by the X-axis and the Y-axis is referred to as the XOY surface, similarly known as yoz surface and zox surface. The three coordinates are divided into eight parts, and each part is called a limit. As shown in the figure "地 示 图", eight is limited to the letter I, II, ..., VIII, including the X-axis, the Y-axis, and the z-axis positive half-axis, in XOY The other three sites on the face are scheduled in the counterclockwise direction, and the limit is based on the limit; in the XOY side, the XOY is adjacent to the set, and then the counterclockwise direction Scheduling sequential is VI, VII, VIII, VIII.

Spatial point Cartial Coordinate

After the spatial right coordinate system O-XYZ, the point of space can be established to establish a correspondence between one ordered array.

Set point M is a point, and the peer m is perpendicular to the plane of the X-axis, the Y-axis, and the Z-axis, respectively. The intersection of three planes and X shafts, Y-axis, and Z axes is set in: P, q, r, points p, q, r, respectively, called point M on the x-axis, Y-axis, and Z-axis. Also set P, Q, R in the X-axis, Y-axis, and the coordinates of the Y-axis and the z-axis are in turn X, Y, Z, and point M determined a ordered array x, y, z. Conversely, if a given array x, y, z can take the point P to be taken on the X-axis, take the point q of the coordinate as Y on the Y-axis, and take the point of the coordinate as Z on the z-axis. R, then P, Q, R are perpendicular to three planes perpendicular to the X-axis, the Y-axis, and the Z-axis, which intersects a point M at which the point m is determined by the ordered array x, y, and z. . In this way, the point M between the space is associated with an orderly array x, y, and z, and the corresponding relationship is established. The ordered array x, y, z is referred to as the coordinates of point M, which is recorded as M (X, Y, Z), where x is called the abscissa, and Y is a vertical coordinate, Z is called a vertical coordinate.

The coordinates of the origin are (0, 0, 0); if the point M is on the X-axis, the coordinates are (x, 0, 0); the same for the point on the Y-axis, the coordinates are (0, y, 0); For points on the Z axis, their coordinates are (0, 0, z); the same, the point on the XOY plane, the coordinates are (x, y, 0); located on the Yoz plane Point, its coordinates are (0, y, z); points located on the Xoz plane, their coordinates are (X, 0, Z). It can be seen that on the coordinate axis, the coordinates and points in the limits, and their coordinates are characterized.

Space two points between the distance

utilized the coordinates of the point, the distance between two points in the space can be obtained.

set a (x ₁ , y ₁ , z ₁ ) and b (x ₂ , y ₂ , z ₂ ) is two points of space, all of which are perpendicular to the plane of the coordinate axis, which are perpendicular to the coordinate axis. A rectangle of AB is diagonal, and its three prisms are

, respectively, due to the distance D between A and B is the length of the rectangular diagonal AB, and

and

are all right triangles, so the campaign is

This is the distance formula between two points of space.

Specially, the distance between the point M (X, Y, Z) and the origin O (0, 0, 0) is