# How is virtual work defined?

Let $$S$$ be a system with $$g$$ degrees of freedom, $$\Sigma_g$$ its configuration space and $$\{q_i\}_{i=1}^g$$ the lagrangian coordinates. If $$P\in\Sigma_g$$, we define a virtual displacement from $$P$$ as follows (using Einstein's convention)

$$\begin{equation} \delta P=\frac{\partial P}{\partial q_i}\delta q_i. \end{equation}$$

As far as I understand $$\left\{\frac{\partial P}{\partial q_i}\right\}_{i=1}^g$$ is a basis for the tangent space in $$P$$, i.e. $$T_P\left(\Sigma_g\right)$$.

What I don't understand is the definition of virtual work for a force $$\boldsymbol{F}$$:

$$\begin{equation} \delta W=\boldsymbol{F}\cdot\delta P. \end{equation}$$

$$\boldsymbol{F}$$ is a vector of ordinary space, while $$\delta P$$ is a vector of tangent space (that has dimension $$g$$). If the definition I wrote above are correct, why is dot product possible?

Please note that I am taking an undergrad course in Classical Mechanics and I've never taken any courses in differential geometry.

Because the $$\delta P$$ is a vector too: the virtual displacement vector that always lies in the tangent space.