Components of wave vector Is 3-dimensional wave vector defined as $$ \tag{1} \mathbf{k}=\frac{2\pi}{\lambda_{x}}\mathbf{\hat{x}}+\frac{2\pi}{\lambda_{y}}\mathbf{\hat{y}}+\frac{2\pi}{\lambda_{z}}\mathbf{\hat{z}}
  ?
$$
If it is, then it's magnitude would be $$ \tag{2} |\mathbf{k}|=2\pi\sqrt{\frac{1}{\lambda_{x}^{2}}+\frac{1}{\lambda_{y}^{2}}+\frac{1}{\lambda_{z}^{2}}}
 .$$
But on the other hand, we know that the magnitude of a wave vector is given by $$ \tag{3} |\mathbf{k}|=\frac{2\pi}{\lambda}.
 $$ So in order for these two to be equivalent it can't be true that $$ \tag{4} \lambda=\sqrt{\lambda_{x}^{2}+\lambda_{y}^{2}+\lambda_{z}^{2}}.$$ 
But doesn't this follow from Pythagoras's theorem so it should be true? I would guess that definition ($1$) is not true but this Wikipedia article about particle in a box and its section 'Higher-dimensional boxes' seems like it uses definition ($1$). Or maybe it doesn't, but then is it not true that $$ \tag{5} \lambda_{x}=\frac{2L_{x}}{n_{x}}
 ?
 $$ If not, could you explain why?
 A: (1) is a definition of $\lambda_x$, $\lambda_y$ and $\lambda_z$, by identification of k components on your base. 
(2) expresses the norm of k. This is Pythagora's theorem.
(3) is a definition of $\lambda$, from which you can calculate its value in terms of $\lambda_x$, $\lambda_y$ and $\lambda_z$.
There is no need for Pythagora's theorem to hold  in any arbitrary space like $\lambda$'s space. Indeed, you've just proved that it didn't here. It is rather the contrary: you create a Hilbert space by defining perpendicularity / dot product through the extension of Pythagora's theorem.
Addendum after discussion with OP in the comment section
It is not easy to define a natural length here. You have a 3D pattern which is periodic, so it lies in a rectangular parallelepiped and reproduces itself every $\lambda_x$ on $x$ axis, $_$ on $y$ axis and $_$ on $z$ axis. Rectangular parallelepiped's diagonal is not an "obvious" wavelength. Taking a kind of quadratic mean (up to a ${1 \over \sqrt{3}}$ factor) makes more or less sense. And, as @BySymmetry said, if the wave is 1D or 2D, using Pythagora would give $\lambda = \infty$ whatever the wave, which wouldn't be very useful.
A: Equation (4) is clearly not true. Consider a simple plane wave propagating in the x direction. Then clearly we should have $\lambda = \lambda_x$. However $\lambda_y$ is infinite, so equation (4) does not work. (If you don't like the idea that $\lambda_y$ should be considered infinite, you can instead consider a wave making a small angle to the x axis, so $\lambda$ is approximately $\lambda_x$, but $\lambda_y$ is very large.) 
A: As you seem to have found out, eq. 1 is wrong, because the wavelengh is not a vector. It is a scalar.
$k=2\pi/\lambda$
for the case of a one dimensional wave.
For a higher-dimensional one, you have to consider the normal vector, that is, the unit vector that points in the direction of propagation.
$\hat{n}=(n_x, n_y, n_z)$
And it is a unit vector, so that $n_x^2+n_y^2+n_z^2=1$
Then, the wave vector is 
$$\vec{k}=\frac{2\pi}{\lambda} (n_x, n_y, n_z)$$
So of course its modulus remains $2\pi/\lambda$.
A: Equation (3) is only true for a one dimensional wave. For three dimensional waves, the relation between wave number and wave is clearly shown in Eq. (1):
$K_x=\frac{2\pi}{\lambda_x}$
and so on.
Consider a 2D wave in cartesian coordinate system. Let the wave vector be $\vec{k}=(k_x,k_y)$. The wave vector is perpendicular to the direction of motion of waves, by definition. Let's say you observe these waves and you'd want to count number of waves per unit length in a particular direction. If you stand on the $x$-axis, you'd count $k_x$ number of waves, and $k_y$ if you stand on the $y$-axis. If you look at the waves along the wave vector direction, you'd see $\lvert k \rvert$ number of waves. Hence the formula $\lvert k \rvert = \frac{2\pi}{\lambda}$. The wavelength has to be measured along the direction of motion of the waves. Otherwise you'd see apparent wave length like $\lambda_x$, for which we have $k_x = \frac{2\pi}{\lambda_x}$ and so on.
