In the Book Analytical Mechanics by Fasano and Marmi on page 128 while discussing holonomic constrains, the book makes the following assertion:
Since the vectors $(\partial X/\partial q_k)_{k=1,...,l}$ are a basis of the tangent space $(T_X \mathcal V(t))$ in a fixed system of Lagrangian coordinates $(q_1,...,q_l)$, the projection onto $(T_X \mathcal V(t))$ of a vector $Z$ is uniquely determined by the components
$$ Z_{\Theta,k}=Z\cdot\frac{\partial X}{\partial q_k} $$
I'm not quite sure what to make of this. If the basis elements were orthonormal then these would be the exact components of the projection. If they were orthogonal they wouldnt necessarily be the components, but would uniquely determine them. Same holds if the dot product was defined with respect to the given basis, but however since we're talking about manifolds I'm assuming that would require further work showing the existence. In my mind the only remaining option is that independent of the scalar product, these values do uniquely determine the coordinates of the projection, this however I was not able to show and think is false.
Dot products with any basis will determine a vector, just in an awkward way. Let $\xi_1,\dots,\xi_n$ is a basis for an inner product space $V$, and set $g_{ij}= \xi_i\cdot\xi_j$. Then, if we write $Z=\sum\limits_{i=1}^n a^i\xi_i$, we have $$b_j=Z\cdot\xi_j = \sum_{i=1}^n a^i\xi_i\cdot\xi_j = \sum_{i=1}^n a^ig_{ij}.$$ As is standard, we denote by $g^{jk}$ the inverse matrix of $(g_{ij})$, so that $\sum\limits_{j=1}^n g_{ij}g^{jk} = \delta_i^k$. Then from $b_j=\sum\limits_{i=1}^n a^ig_{ij}$ we deduce that $$\sum_{j=1}^n b_jg^{jk} = \sum_{j=1}^n\left(\sum_{i=1}^n a^i g_{ij}\right) g^{jk} = \sum_{i=1}^n a^i\left(\sum_{j=1}^n g_{ij} g^{jk}\right) = \sum_{i=1}^n a^i\delta_i^k = a^k.$$