Wavelength scalar/vector ambiguity? I have seen the de Broglie relation for wavelength written as
$$\lambda = \frac{h}{\gamma(\vec{v})m_o\vec{v}}$$
I know wave propagation in the one-dimensional case makes wavelength a scalar, as the velocity term will only have one component. However, if the velocity has say $n$ coordinates, what is the wavelength now? Is it an $n$-dimensional vector thats a Hadamard inverse of relativistic momentum times $h$? Or is magnitude taken of momentum first to produce a scalar? Intuition leads me to believe the latter...
 A: You are right that the wavelength $\lambda$ is not a vector,
and therefore the equation
$$\lambda = \frac{h}{\gamma(\vec{v})m_0\vec{v}} \tag{1}$$
didn't make sense as a vector equation.
But the reciprocal wavelength ($1/\lambda$) can be regarded a vector.
To understand this consider the image below, showing a plane wave.
(I show only in 2 dimensions, because it is easier
to draw. But you can do the same reasoning also in 3 dimensions.)

The wave has wavelength $\lambda$.
Along the $x$-direction it has a wavelength $\lambda_x$, and likewise,
along the $y$-direction a wavelength $\lambda_y$.
Notice also that $\lambda_x \ge \lambda$ and $\lambda_y \ge \lambda$.
It can be shown that the 3 components
$\left(1/\lambda_x,1/\lambda_y,1/\lambda_z\right)$
have all properties of a vector, including the
transformation behavior when rotating the $xyz$ coordinate system.
This vector is called the wave vector or wave number.
$$\vec{k}
= \begin{pmatrix} k_x \\ k_y \\ k_z \end{pmatrix}
= \begin{pmatrix} 1/\lambda_x \\ 1/\lambda_y \\ 1/\lambda_z \end{pmatrix} \tag{2}$$
(There are different conventions in use for this definition,
either with or without an additional factor of $2\pi$.
For simplicity here I choose the one without $2\pi$.)
Using this definition (2) you can write the de Broglie relation,
instead of equation (1), as a true vector equation.
$$ h\vec{k} = \gamma(\vec{v})m_0\vec{v} \tag{3}$$
A: The equation written is wrong, in a sense. In this form itself, the denominator must be a modulus because the wavelength is a scalar, but as you may remember the wavenumber k is a vector which gives us the direction of the wave and it's magnitude gives us the number of complete cycles in one unit of distance.
Hence, if you write the above equation in terms of the wavenumber vector, it's fine and good, otherwise, yeah, it's wrong because Wavelength, by definition is a scalar.
