Exploring Quantum Physics - Basic Wave Equation

In this post I would like to summarize what I understand about the wave equation.

The wave equation look like this $u(x, t) = A \sin ( k x - \omega t + \phi)$.

The number $k$ is the wave vector. This confused me because it looks like a scalar to me.

Suppose time freezes, if we advance $x$ by $\frac{2\pi}{k}$, we get $u(x + \frac{2\pi}{k}, t) = A \sin ( k (x + \frac{2\pi}{k}) - \omega t + \phi) = A \sin ( k x + 2\pi - \omega t + \phi) = A \sin ( k x - \omega t + \phi)$. Therefore, $\frac{2\pi}{k}$ is the wave length.

Switching to the time scale, if position freeze and we advance time by $\frac{2\pi}{\omega}$. It is apparent that using the same trick we get the same value. This means $\frac{2\pi}{\omega}$ is the period, $\frac{\omega}{2\pi}$ is the frequency, and $\omega$ is the angular frequency, nothing special about this.

To advance one wavelength, one need one period of time. Therefore the speed of the wave is $v = \frac{\lambda}{T} = f\lambda = \frac{\omega}{2\pi}\frac{2\pi}{k} = \frac{\omega}{k}$.