Similar to spatial analytic geometry, in order to determine the position of any point in space, it is necessary to introduce the coordinate system into space, and the most commonly used coordinate system is the spatial rectangular coordinate system.

Choose a point O in the space at will, and make the intersection O into three mutually perpendicular number axes Ox, Oy and Oz, all of which take O as the origin and have the same length unit. These three axes are called X axis (horizontal axis), Y axis (vertical axis) and Z axis (vertical axis) respectively, and are collectively referred to as coordinate axes. Their positive direction conforms to the right-handed rule, that is, the right hand holds the Z axis, when the positive direction of the X axis of the four fingers of the right hand is

When the angle turns to the positive direction of Y axis, the thumb points to the positive direction of Z axis. In this way, a spatial rectangular coordinate system is formed, which is called spatial rectangular coordinate system O-xyz. The fixed point o is called the origin of the coordinate system. Corresponding to this is the left-handed rectangular coordinate system. The right-handed rectangular coordinate system is commonly used in general mathematics, but it is different in other disciplines because of its convenient application.

Any two coordinate axes determine a plane, so that three mutually perpendicular planes can be determined, which are collectively called coordinate planes. The coordinate plane defined by X axis and Y axis is called xOy plane, and there are also yOz plane and zOx plane. Three coordinate planes divide the space into eight parts, and each part is called a divination limit. As shown in the picture on the right, the eight diagrams use letters I, II, ... and VIII respectively, in which the first hexagram contains the positive semi-axis of the X-axis, Y-axis and Z-axis, and the other three hexagrams on the xOy plane are arranged counterclockwise, followed by the second, third and fourth hexagrams. Below the xOy plane, the hexagram limit adjacent to the first hexagram limit is the fifth hexagram limit, and then it is also arranged counterclockwise as the sixth hexagram limit, the seventh hexagram limit and the eighth hexagram limit.

After taking the rectangular coordinate system O-xyz, we can establish a one-to-one correspondence between points in space and ordered arrays.

Let the point m be a point in space, and the passing point m be planes perpendicular to the X-axis, Y-axis and Z-axis respectively. Let the intersections of three planes with X-axis, Y-axis and Z-axis be P, Q and R in turn, and the points P, Q and R are called the projections of point M on X-axis, Y-axis and Z-axis respectively. Let the coordinates of points P, Q and R on the X-axis, Y-axis and Z-axis be X, Y and Z in turn, then point M determines an ordered array X, Y, Z ... On the other hand, if an ordered array X, Y and Z is given, we can take point P on the X-axis, point Q on the Y-axis and point R on the Z-axis. The point M is determined by the ordered array X, Y, Z, so that the point M in space corresponds to the ordered array X, Y, Z one by one, and the ordered array X, Y, Z is called the coordinate of the point M, which is denoted as M(x, Y, Z), where X is called the abscissa, Y is called the ordinate, and Z is called the ordinate.

The coordinate of the origin is (0,0,0); If point M is on the X axis, its coordinate is (x, 0, 0); Similarly, for the point on the Y axis, its coordinate is (0, y, 0); For the point on the Z axis, its coordinate is (0,0,z); Similarly, the coordinate of a point on the xOy plane is (x, y, 0); A point on the yOz plane with coordinates (0, y, z); A point on the xOz plane with coordinates (x, 0, z). It can be seen that the coordinates of points located on the coordinate axis, the coordinate plane and within each hexagonal boundary have their own characteristics.

Using the coordinates of points, we can find the distance between two points in space.

Let A(x 1, y 1, z 1) and B(x2, y2, z2) be two points in space, and make three planes perpendicular to the coordinate axis through a and b respectively. These six planes enclose a cuboid with AB as the diagonal, and the lengths of its three sides are respectively

Because the distance d between a and b is the length of the diagonal AB of the cuboid, and

and

They are all right-angled triangles, so they are obtained by Pythagorean theorem.

This is the distance formula between two points in space.

In particular, the distance between the point M(x, y, z) in space and the origin O(x, y, z) is

I hope it can help you solve the problem.