Math_Calculus_07_ vector algebra and analytic geometry

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Chapter VII of the vector and Analytic Geometry

§7.1 spatial rectangular coordinate system

7.1.1 Cartesian coordinate system of spatial points

Space through a point O, perpendicular to each other as three, have the same length in units of the number of axes which are referred to as x-axis, y-axis and z-axis, and such that they conform to the forward right-hand rule, i.e., the right hand holding the z-axis, when four fingers of the right hand from the positive x-axis, at an angle steering positive y-axis, z-axis thumb points forward, as shown in FIG 7-1.

Pointing arrow in FIG represents the x-axis, y-axis, the positive z-axis. Such three axes on the formation of a spatial rectangular coordinate system. Is called the coordinate origin point O (or origin).

Any two of the three coordinate axes in a plane can be determined, so that three planes collectively fix a coordinate plane. Coordinate plane by the x and y axes is called the determined xOy plane, the other two by the y-axis and z-axis of the coordinate plane and the z-axis and the x-axis is determined, and are called zOx yOz plane surface.

Three spatial coordinate plane is divided into eight parts, each part is called octants. Containing the x-axis, the y and z axes Gua positive axis is called the first octant limit, the other second, third, fourth octants, above the xOy plane, determining counterclockwise. Gossip fifth to second threshold, below the xOy plane, the counterclockwise direction is determined by the fifth octant below the first octant, the eight octants by the letters I, II, III, IV, V , VI, VII, VIII representation.

7.1.2 coordinate point

Let M be a known spatial points. We had three point M for each plane perpendicular to the x axis, y axis and z-axis, which x-axis, y-axis, z-axis were intersection P, Q, R & lt (FIG. 7-2), in which three x axis, y axis, z axis coordinates were x, y, z. Thus one o'clock M space uniquely determined on an ordered array of x, y, z; conversely, it is known an ordered array x, y, z, we can take the point P of coordinates x in the x-axis, in taken as y-axis coordinate y of the point Q, taken in the z-axis coordinate z of the point R, then R and P Q respectively, for the x-axis, y-axis and a plane perpendicular to the z-axis. The three M is the intersection of the vertical plane of an ordered array of single point x, y, z determined. Thus, it is established one correspondence between the spatial points and ordered array of M x, y, z. This set of numbers x, y, z coordinates of the point M is called, and in turn, said x, y and z of the point M on the abscissa, ordinate and the vertical coordinate. Coordinates x, y, z of the point M is generally referred to as M (x, y, z) .

A point coordinate plane the coordinate axis has its characteristics, specifically:

If the point M in the yOz plane, the coordinates of the point M in the X = 0 ; point M in zOx surface, the coordinate point M of Y = 0 ; point M in xOy plane, the coordinates of the point M the Z = 0 . I.e., the coordinates of characteristic points in the coordinate plane is: Which point in the coordinate plane root, which coordinate point is zero. The equation for the plane xoy , xoy plane of the points is .
If the point M in the x -axis, then the coordinates of the point M of y = z = 0 ; y-axis at point M, at the coordinate point M z = x = 0 ; point M in z -axis, then the coordinates of the point M in the X = Y = 0 . I.e., the coordinates of characteristic points in the coordinate axis are: the point where the root of the coordinate axes, the coordinates of which non-zero, and the other two coordinate points is zero.
If the point M as the origin, the X = Y = Z = 0 .

7.1.3 distance between two points in space

Set , for the two points in space, with the two coordinates in order to express the distance d between them, we had a M _{. 1} , M ₂ each for three separate plane perpendicular to the three axes. Plane surrounded by a six to M _{. 1} M ₂ is a diagonal rectangular (FIG. 7-3).