§ 4 spatial Linear Equation

§ 4 spatial Linear Equation

General equation of a Spatial Straight Line:

The Spatial Straight line can be regarded as the intersection of two planes. In fact, if the plane and equation of the two intersections are

:

:

Then, the coordinates of any point on the Spatial Straight Line satisfy both the plane equations, that is, the equations must be satisfied.

(3.4-1)

In turn, if a point is not on a straight line, it cannot be on both the plane and the plane, so its coordinates do not meet the equations (3.4-1 ).

Therefore, it can be expressed by equations (3.4-1). equations (3.4-1) are called General Equations of spatial straight lines.

Generally, there are more than one plane that goes through a spatial always-linear line, so you only need to select two of them in an infinite number of planes and combine their equations, the equation of the Spatial Straight Line is obtained.

 

Symmetric equations and parameter equations of two Spatial Straight Lines:

If a non-zero vector is parallel to a known straight line, this vector is called the direction vector of the straight line. Obviously, any vector on the straight line is parallel to the direction vector of the straight line.

We know that a single point of space can be used and can only be used as a straight line parallel to a known straight line. Therefore, when a point on a straight line is given with one of its direction vectors, the location of the Spatial Straight Line is completely determined.

Next, let's establish this linear equation.

 

3-5, set to any point on the straight line, then,

While

,

So. (3.4-2)

In turn, if the point is not in a straight line, it is not parallel, and thus the formula (2) is not true.

Therefore, equations (3.4-2) are linear equations.Symmetric Equation.

The coordinates of any direction vector of a straight line are called a group of direction numbers of the straight line, and its direction cosine is called the direction cosine of the straight line.

As shown in

Then (3.4-3)

Equations (3.4-3) are called linearParameter equation.

Example 1Use Symmetric Equation and Its Parameter Equation to represent a straight line

SolutionFirst, find out a point in this line, for example, get the result

To solve this binary system,

So we can get a point in a straight line.

Find another direction vector of the line, because the intersection of the two planes and the normal vector of the two planes

Both vertical and desirable

,

Therefore, the Symmetric Equation of the given straight line is

;

The linear parameter equation is