Lecture 2.2. Straight line in space Different forms of the equation of a straight line in space

Vector form. Let be the straight line passing through the point and parallel to the nonzero vector (called a direction vector). It is clear that consists precisely of those points for which the vector is collinear to , or . (1)

Here , , t is a parameter and . Equation (1) is called the

vector form of the equation of straight line L.

Parametric form. When we separate the components in vector equation (1), we obtain the parametric equations of a t line.

. (2)

Standard form

We may eliminate t from the equations in (2) to obtain the standard form of the equation of a straight line

. (3)

The numbers l, n and m are the coordinates of the direction vector and they are called direction numbers of the line L.

Equation of a line passing through two given points

Suppose we are given two points and . Vector is collinear to the line passing through these points. It follows that vector

is a direction vector to the line. Thus we have the standard equation of a line

.

Line of interception of two non parallel planes

Every equation of first order represents a plane, and two non-parallel planes P₁ and P₂ intersect in a straight line L. Hence the locus of two simultaneous equations of first order

(4)

is a straight line. The line L lies in each plane and will be perpendicular to both the normal to P₁ and the normal to P₂, showing its direction must be along the vector of cross–product . Therefore we can take the vector as a direction vector of L . Taking into account the formula for the cross–product we obtain

. (5)

To find the coordinates of any point on the line L it is sufficient to solve linear algebraic system (4). Then using the point and direction vector from (5) we can write down the standard form of equation of the line L.

Typical problems on a line in space

1. The angle between two straight lines in space is the angle between their direction vectors and : .

Therefore two lines are parallel if and they are perpendicular if .

2. The projection of a straight line to the coordinates planes

Standard form (3) of the equation of a line L in space is a double proportion and can be considered as a system of two equations of planes in form (4). For instance, P₁ and P₂:

Every proportion is the equation (3) is the equation of the plane containing the line L and perpendicular to the corresponding coordinate plane, so it can be considered as the equation of the projection of the line L on the plane Oxy, Oxz, and Oyz correspondently:

(Oxy) ; (Oxz) ; (Oyz) .

3. The distance from a point to a straight line in space

Let there be given a line L passing through a point and having a direction vector . Then the distance d from any arbitrary point to the line L may be found as the high of the parallelogram formed on the vectors and by the following formula:

. (6)

4. Two lines lie in the same plane if one of the following conditions are true:

1. they are parallel (or and are collinear);

2. they are concurrent or the distance d calculated by formula (7) is zero.

5. The point of intersection of two lines in space may be found as solution of their equations considered simultaneously. It is convenient to use the parametrical form of the equations.