A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation.
The wave vector is most useful for generalizing the equation of a single wave into a description of a family of waves. As long as the family of waves all travels in the same direction and with the same wavelength, a single wave vector is valid for the entire family. The most common case of a family of waves that meets these requirements is the plane wave, in which the family of waves is also coherent, i.e. all the waves have the same phase.
For example, a common representation of a single wave at a single point in space is:
where A is the amplitude, ω is the angular frequency, and φ is the starting phase of the wave (the independent variable t is time).
In order to generalize the equation to all points in the one-dimensional space of the direction of propagation, we add in an additional phase offset term:
where k is the wavenumber (2π/λ) and the new independent variable z is the distance along the wave.
Now, as long as we are dealing with a simple family of waves, with identical direction, wavelength, and phase (i.e. a plane wave), we can easily extend the formula by substituting the wave vector k for the wavenumber k, and the location in space vector r for the variable z: