§ 5 frame and coordinates

§ 5 frame and coordinates

 

ICartesian coordinates of spatial points:

The Cartesian coordinate system of the plane establishes a one-to-one correspondence between points on the plane and a pair of Ordered arrays, and communicates the research on the plane graphics and numbers.

In order to communicate the research of spatial graphics and numbers, we use a method similar to the plane analytic ry, which is achieved by introducing the spatial Cartesian coordinate system.

1,Space Cartesian coordinate system

When the space passes through a certain point, it is used as three vertical numbers. They are thought of as the origin and generally have the same length Unit. These three axes are called the axis (horizontal axis) and the axis (vertical axis) respectively), axis (vertical axis), and collectively referred to as the axis.

Usually, the axes and axes are configured on the horizontal plane, while the axes are vertical lines, and their positive direction must comply with the right-hand rule:

(Fig. 1.13)

Hold the axis in the right hand. When the four fingers in the right hand are in the forward direction of the axis from the angle, the point of the thumb is in the forward direction of the axis.

The three coordinate axes form a space Cartesian coordinate system. The point is called the coordinate origin..

Note: To make the space Cartesian coordinate system more stereoscopic, usually draw the angle between the axis and the axis to the left and right. Of course, their actual angle is still.

2,Coordinate Plane and limit

Any two of the three axes can determine a plane, so that the three planes are collectively referred to as the coordinate plane.

The coordinate plane determined by the axis and the axis is called the plane. In addition, the plane and the plane are also called.

The Three Coordinate Planes divide the space into eight parts, which are called limit.

(Fig. 1.14)

3,Cartesian coordinates of spatial points

After obtaining the Cartesian coordinate system of a space, we can establish the correspondence between the spatial point and the ordered array.

It is a known point of space. The point of Pass is a plane perpendicular to the axis, the axis, and the axis. The point of intersection between them and the axis, the axis, and the axis is in sequence, these three points are in sequence in the axes, axes, and axes. Therefore, space points uniquely determine an ordered array, which is called the coordinates of points.

It is called in sequence, which is the abscissa of a point.,Vertical and vertical coordinates, recorded.

(Fig. 1.15)

In turn, if an ordered array is known, we can take the coordinates of the point on the axis, take the coordinates of the point on the axis, and take the coordinates of the point on the axis, then, and are used as the vertical plane of the axis, axis, and axis respectively. The intersection of these three planes is the spatial point coordinate with an ordered array.

In this way, through the space Cartesian coordinate system, we establish a one-to-one correspondence between the space points and the ordered array.

Definition 1We call the ordered array above the coordinates of the vertex in this coordinate system.

Distance between two points in the second Space

Theorem 1The distance between two points is

(1.5-1)

Certificate
And each of them is perpendicular to the plane of the three coordinate axes. These six planes are enclosed in a cube with a diagonal line,

 

(Fig. 1.16)

Is a right triangle, so

,

Because it is a right triangle

,

Thus;

However,

,

,

Therefore.

In particular, the distance between the point and the coordinate origin is

.

3.Coordinate of spatial Vectors

Definition 2It is a unit vector in the same direction as the coordinate axis. It has a unique set of real numbers for any space vector, so we call this set of ordered real numbers the coordinates of the vectors in this coordinate system, recorded as or.

Theorem 2

If the constant coordinate of the vector is, then the coordinate of the vector is

. (1.5-2)

CertificateDefinition of point and vector Coordinate

,

So

=.

By definition.

Theorem 3The component of the two vectors is equal to the sum of the components of the two vectors.

CertificateSet, then

= +

=,

So. (1.5-3)

Likewise, we can prove the following two theorems:

Theorem 4

Set, then.

Theorem 5If this parameter is set, the wildcard must meet the following conditions:

. (1.5-4)

Theorem 6

Three non-zero vectors. The required and sufficient conditions for the common surfaces are:

. (1.5-5)

CertificateBecause there is no common surface, there is a real number with not all 0 to make

,

Therefore

Because not all values are 0

.