Trajectory By Christoffel Symbols

So, basically Christoffel Symbols gives us the components of the vector that represents the rate of change of the basis vectors in a manifold (surface).

And this equation below tells the rate of change of the vector components of a velocity vector summarized by a given object in a manifold where the Christoffel Symbols != 0,

$$frac{dv^a}{dt} = -Gamma_{mn}^a v^m v^n$$

Let me be clear with simple terms, $v^m$ and $v^n$ are vector components respectively such that $v^1$ is the x-component, $v^2$ is the y-component of the velocity vector simultaneously.
And $Gamma$ with indices is the Christoffel Symbol, if you are interested with its derivation I will leave a link below for a video that explains step by step derivation and explanation.

My question is that will this equation hold true for in 3 spatial dimensions where the velocity vector has three components in each direction respectively?

Or it will take the form,

$$frac{dv^a}{dt} = -Gamma_{m n}^a v^m v^nv^{i}$$

And $a, m,n,i$ can take value $1,2,3$.
The link to the video:

• ScienceClic English. "The Maths of General Relativity (3/8) – Geodesics
  ", YouTube, Dec 8, 2020.