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 М0 and М to the origin, and find the coordinates of the radius-vectors , . It is seen from the figure, that . If the point М 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 M0(x0,y0,z0) and the direction vector . Let us write the left-hand 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 .
The equation of the straight line passing through two given points. Suppose given two points and in space. z M2 M1
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