STRAIGHT LINES IN SPACE
The Vector, Parametric, Canonical, and General Equations of a Straight Line
The position of a straight line in space is determined by a point on this line and a vector parallel to the line. Let us write an
equation of such a line in space.
z
0 y
x
To this end, we take an arbitrary point on the line, join Ì_{0} and Ì to the origin, and find the coordinates of the radiusvectors
,
.
It is seen from the figure, that .
If the point Ì belongs to the straight line, then the vectors and are collinear.
Consequently, these vectors meet the collinearity condition
,
where t is a parameter.
Let us write the collinearity condition in the form
; (*)
equation (*) is the vector equation of the given line.
Suppose given the coordinates of the point M_{0}(x_{0},y_{0},z_{0}) and the direction vector . Let us write the lefthand side of equation (*) in the vector form
the direction vector is
.
Let us represent equation (*) in the form
.
Equating the respective coefficients of the unit vectors on the right and left sides, we obtain parametric equations of the straight line:
or (27)
Eliminating the parameter t, we obtain the canonical equations of a straight line:
. (28)
Example. Write the canonical equations of the straight line passing through the point parallel to the vector . We compose the canonical equation by formula (28):
.
Equating each fraction to a parameter t, we obtain the parametric equations of the line:
The general equation of a straight line in space. Since a straight line in space is represented as the intersection of two planes, the general equation of a straight line in space has the form of a system
where the first and the second equations are the equations of the corresponding planes.
It is always possible to transform the general equation of a straight line into a canonical equation and vice versa.
Since the direction of is perpendicular to those of the vectors and , it follows that
,
i.e., the canonical equation is
.
The angle between straight lines in space. The parallelism and perpendicularity conditions for straight lines. Let us find the angle between intersecting right lines given by their canonical equations
; .
The angle between these two lines is equal to the angle between their direction vectors
; ,
i.e.,
.
The parallelism and perpendicularity conditions for right lines coincide with the collinearity and perpendicularity conditions of their direction vectors and .
If straight lines are perpendicular, then , i.e., , and the perpendicularity condition is
.
If straight lines are parallel, then the vector is collinear to , i.e., their coordinates are proportional, and the proportionality condition is
.
Date: 20150102; view: 1137
