nLab direction of a vector

This entry is about the concepts of direction of a vector and of a line in linear algebra/analytic geometry. For the concept in order theory see at direction.

Idea

In vector/affine geometry the parallel? vectors with the same orientation form an equivalence class. The same holds for oriented lines and even higher dimensional oriented subspaces. The equivalence class of a vector (or other object) is called its direction. (In the case of higher dimension we may say “$n$-direction”; see also bivector.) If the vector space is normed, then each such equivalence class has a canonical unit vector representative, called a direction vector.

If we neglect the orientation we get a bigger equivalence class, called the unoriented direction. Unoriented directions have a representing unit vector only up to sign.

Definition

Oriented direction

Definition

For $V$ a real vector space, or more generally a vector space over an ordered field, the direction of a non-zero vector $v \in V$ is its equivalence class under the multiplication with positive scalars.

If $V$ is a normed vector space (such as an inner product space with the induced norm $v\mapsto \sqrt{\langle v, v\rangle}$), the direction of $v$ is canonically represented by the unit norm vector obtained by multiplying $v$ with the inverse of its norm

$$\tfrac{1}{\vert v\vert} v \;\in \; S(V) \subset V \,,$$

which may be regarded as an element of the unit sphere $S(V)$ of $V$.

(e.g. MathWorld)

Equivalently:

Recall that a nonzero vector $v$ is parallel to a linear subspace $W$ of a vector space $V$ if $v$ has a representative of the form $\vec{A B}$ (i.e. $v = B - A$) with $A,B\in W$. When $W$ is a line, we say that $v$ furthermore has the same orientation as $W$ if $\vec{A B}$ points (in the sense of def. ) in the positive direction along $W$.

Definition

The direction of a non-zero vector $v \in V$ is the equivalence class of oriented lines parallel to $v$ and having the same orientation.

Unoriented direction

Similarly, the unoriented direction of a non-zero vector in any vector space is its equivalence class under multiplication by non-zero elements of the ground field, hence the element it represents in the corresponding projective space $P V$. Unlike oriented direction, this makes sense over an arbitrary field.

The unit direction vector can be attached not only to a (direction of) a vector but also to a direction of lines per se (see Wikipedia, Direction vector).

J. Lelong-Ferrand defines the direction of an affine subspace of an affine space as its orbit under the action of the translation group. This agrees with the parallelness in Choquet where two affine subspaces of the same dimension are parallel if their translation group of vectors (as a subgroup of the group of vectors of translations of the entire space) is the same.

Examples

Example

(wave fronts)

In microlocal analysis the wave front set of a distribution records the direction (def. ) of all those wave vectors along which, locally, the Fourier transform of the distribution is not a rapidly decaying function.

Related concepts

• slope of a line, tangent

• norm, length, absolute value

• wave vector, orientation, bivector

References

• MathWorld, Unit vector

• Gustave Choquet, Geometry in a modern setting, Hermann 1969

• K. Horvatić, Linear algebra (in Croatian), 2004

• Jacquelline Lelong-Ferrand, Les fondements de la géométrie, Presses Universitaires de France 1985.