Lines and Planes in 3D Space
Equation of a line in 3D Space
As we know, we can uniquely determine a line in 3D space by specifying a point on the line and the direction of the line. Suppose a line passing through the point and is in the same direction as the vector (this non-zero vector is called the direction vector of ). Let be any point on line and . We have
i.e. the position vector of .
As is parallel to , for some real number . Therefore, we have
The parametric equation of line is as follows:
where is any real number.
The line is illustrated in the applet below.
Remark: The parametric equation of a line is not unique i.e. different parametric equations can represent the same line. For example, you can choose another point on the line or another direction vector that is a scalar multiple of .
Two lines in 3D space are non-parallel if their direction vectors are non-parallel i.e. one is not a scalar multiple of another. In 2D space, any two non-parallel lines must intersect each other. However, this is generally not true in 3D space, as shown in the following example:
Example:
Let be the line passing through in the direction .
Let be the line passing through in the direction .
Show that they are non-parallel lines that do not intersect each other.
Solution: Obviously, and are not parallel vectors. Therefore line and are non-parallel.
The following are their parametric equations:
for any real number
for any real number
Assume there exists an intersection point between the two lines i.e. there exists and such that they satisfy the above two parametric equation simultaneously. We should be able to solve the following system of equations:
However, it can easily be shown that the above system of equations has no solution! Hence, the two lines never intersect each other.