Direction (geometry)

In geometry, direction is a fundamental concept used to describe how objects point in space. It is also known as spatial direction, vector direction, or relative direction. A direction is the shared characteristic of all rays that coincide when they are translated to a common endpoint. This means if you slide rays across a surface without turning them, they can overlap perfectly. This concept also applies to vectors, such as the relative position between two points. Two vectors share a direction if they can be made equal by scaling them by a positive scalar multiplier.

When two vectors share the same direction, they are called codirectional or equidirectional. If these codirectional line segments also have the exact same length, they are called equipollent. It is important to note that equipollent segments are not always coincident. For example, a single direction can be evaluated at different starting positions. This defines different unit directed line segments, which are viewed as bound vectors rather than free vectors. Furthermore, two rays or oriented line segments that are colinear do not necessarily share the same direction. Colinear means they share the same supporting line, but they might point in different ways.

Mathematicians use several methods to represent a direction numerically. One common method is using a unit vector. You create a unit vector by dividing a given vector by its own length. Another way to represent direction is by using a point on a circle or a sphere. If a ray starts at the center of a sphere and moves outward, its direction is defined by where it intersects the sphere's surface. The tips of unit vectors that all start from a single origin point will lie on this unit sphere.

In a two-dimensional plane, a direction can be represented by a single angle. This angle is measured from a specific reference direction. This is known as the angular component of polar coordinates. In three-dimensional space, the representation becomes more complex. You must use two different angles to define a direction. These are the polar angle, which is relative to a fixed polar axis, and the azimuthal angle, which is measured about that polar axis. These are the angular components found in spherical coordinates.

Another way to specify an arbitrary direction is through a Cartesian coordinate system. This system relies on mutually orthogonal coordinate axes, which are axes that meet at right angles. You can also represent a direction numerically using direction cosines. This is a list of the cosines of the angles associated with the direction. These values are mathematically equivalent to the Cartesian coordinates of the corresponding unit vector. These tools allow scientists to describe the orientation of complex objects in physical space, such as through an axis-angle representation.

Directions can also be compared to one another to see how they relate. Two directions are considered opposite if their unit vectors are additive inverses. On a sphere, these opposite directions are represented by antipodal points. Antipodal points sit at the two opposite ends of a common diameter. When you combine a direction with its opposite, you form an undirected line. This is sometimes called a direction line or an orientation. For instance, the east-west orientation supports both the east direction and the west direction. Two directions are parallel if they can lie on the same straight line without being rotated. Parallel directions are either codirectional or they are opposite.

We can also classify the relationship between two directions based on the angle they form. If the directions form an angle smaller than a right angle, they are called acute directions. If they form an angle greater than a right angle, they are called obtuse directions. Mathematically, this can be determined by their scalar product or scalar projection. Acute directions have a positive scalar product, while obtuse directions have a negative scalar product. Finally, non-oriented straight lines can also have a direction. This direction is the common characteristic of all parallel lines. In a two-dimensional plane, the direction of a non-oriented line is represented by its slope relative to a reference axis.