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Why does this definition of generalized forces work?

I don't understand an equation in the discussion of Lagrangian Mechanics in the book _Mathematical Aspects of Classical and Celestial Mechanics_ by Arnold _et al._ (section 1.2.1).

For context, they are considering a set of $n$ particles. The set of functions $\mathbf{r}_i:\mathbb{R}\to\mathbb{R}^3$ give the position of each particle as a function of time. The particles are constrained to a manifold $M$ embedded in $\mathbb{R}^{3n}$ so that $\mathbf{r}=(\mathbf{r}_1(t),...,\mathbf{r}_n(t))\in M$ for all $t$.

They say "let $q=(q_1,...,q_k)$ be local coordinates on $M$." I'm not sure what exactly they mean by this. They treat $q$ as a function of $t$, so maybe they mean $q=\varphi\circ\mathbf{r}$, where $\varphi:U\subset M\to \mathbb{R}^k$ is a chart that covers the area of interest.

**The Question:** Given vector $\mathbf{F}=(\mathbf{F_1},...,\mathbf{F_n})\in\mathbb{R}^{3n}$, they define the generalized force covectors $Q(q)$ by the equality
$$\sum_{i=1}^n\langle F_i, d\mathbf{r_i}\rangle=\sum_{j=1}^k Q_j dq_j.$$
Is this well-defined? How can we define a covector just by its inner product with a single vector?