A direction vector is a vector that indicates the direction of a line or a plane in a given space.
It does not fix a specific point in space, but only describes the direction in which a line or a plane extends.
1. Direction vector of a line
In analytic geometry, a line in 
or 
can be expressed through a parametric equation of the form:
r(t) =
where:
- r(t) is the position of any point on the line.
is a fixed point on the line (a point the line passes through).- t is a real parameter.
- v is the direction vector, which determines the orientation of the line.
Example: if we have the line:
r(t) = (1,2,3) + t(2, -1, 4)
the direction vector is:
v = (2, -1, 4)
This means that any point on the line can be obtained by adding to the initial point (1,2,3) a multiple of the vector (2, -1, 4), which defines the direction.
2. Direction vector of a plane
In the case of a plane, instead of a single direction vector, we need two direction vectors
v_1 and v_2, since a plane has two independent directions.
This means the plane can be expressed as:
where s and t are real parameters.
3. Properties of direction vectors
- Not unique: any scalar multiple of a direction vector is also a direction vector of the same line.
- Equations: they are used to determine equations of lines and planes.
- Angles and parallelism: in Euclidean geometry, they help find angles between lines and check whether two lines are parallel or coincident.
