Euclidean vector

The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion.

The term vector also has generalizations to higher dimensions, and to more formal approaches with much wider applications.

It has been proven that the two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations.

This coordinate representation of free vectors allows their algebraic features to be expressed in a convenient numerical fashion. For example, the sum of the two (free) vectors (1, 2, 3) and (−2, 0, 4) is the (free) vector

Illustration of tangential and normal components of a vector to a surface.

The choice of a basis does not affect the properties of a vector or its behaviour under transformations.

The following section uses the Cartesian coordinate system with basis vectors

Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors

Two vectors are opposite if they have the same magnitude but opposite direction. So two vectors

Scalar multiplication of a vector by a factor of 3 stretches the vector out.

This happens to be equal to the square root of the dot product, discussed below, of the vector with itself:

The dot product can also be defined as the sum of the products of the components of each vector as

For arbitrary choices of spatial orientation (that is, allowing for left-handed as well as right-handed coordinate systems) the cross product of two vectors is a pseudovector instead of a vector (see below).

The scalar triple product is linear in all three entries and anti-symmetric in the following sense:

With the exception of the cross and triple products, the above formulae generalise to two dimensions and higher dimensions. For example, addition generalises to two dimensions as

A seven-dimensional cross product is similar to the cross product in that its result is a vector orthogonal to the two arguments; there is however no natural way of selecting one of the possible such products.