Let $(M,g)$ be a flat spacetime, $(\gamma:\mathbb{R}\rightarrow M,e)$ an observer (where $e_1(\lambda)$, $e_2(\lambda)$, $e_3(\lambda)$ together with $\dot{\gamma}(\lambda)$ form the basis of the observers frame at $\gamma(\lambda)$) and $\delta:\mathbb{R}\rightarrow M$ a particle.
How does the observer generically observe the speed of $\delta$?
I heard that, in the case where $\gamma(\lambda)=\delta(\lambda)$ for some $\lambda\in\mathbb{R}$, the observer will measure the following velocity vector for the particle:
$$\text{v}_{\delta}:=\sum_{i=1}^3\in^i(\dot{\delta}(\lambda))\cdot e_i$$
where $\in^i$ is the induced dual basis of $T_{\gamma(\lambda)}M$ w.r.t. to the basis $e$.
Say $M=\mathbb{R}^3$ so that we have a visaulizable, 3 dimensional spacetime manifold with two spatial dimensions from the viewpoint of an observer, and
$$g_{ij}=\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{bmatrix}_{ij}$$ so that it is flat. Say we have two worldlines
$$\text{E},\text{P}:\mathbb{R}\rightarrow M,\ \text{E}(\lambda)=(\lambda,0,0),\ \text{P}(\lambda)=(\lambda,1,\lambda\cdot v)$$
for some $v\in\mathbb{R}^+$. Then how will an observer following $\text{E}$, given any frame, perceive the velocity of $\text{P}$?
My first approach went very much wrong. To show effort let me summarize: We can view the two light rays $L_1$ and $L_2$ between $\text{P}$ and $\text{E}$ - at two pairs of $\lambda$-values, $(\lambda_1^P,\lambda_2^P)$, $(\lambda_1^E,\lambda_2^E)$, for $\text{P}$ and $\text{E}$ such that they seperate $\text{E}$ by a time interval of length 1 - as vectors in $\text{E}$s frame $(T_{\text{E}(\lambda_{1/2}^E)}M)$ at this observer's parameter values ($\text{E}$ will 'see' $\text{P}$ at these points in (his/her) time via the world lines of the two lightrays). We can parallely transport $L_1$ to the frame at parameter value $\lambda^E_2$, where $L_2$ lives, and subtract the two. Maybe if we measure $L_2-L'_1$s length with $g$ this will give us something akin to the velocity $\text{E}$ would measure of $\text{P}$? But that turns out not to be the case.
I read online that one can calculate the subjective distance in special relativity (using the tools of the manifold-spacetime) as follows: Measure the travel time observed by $\text{E}$ of a lightray sent out at some $\text{E}(\lambda)$, being reflected by $\text{P}$s worldline and consequently crossing $\text{E}$ again. Multiplying this measured time by the speed of light divided by 2 will give the perceived distance between $\text{E}$ and $\text{P}$. But in our example of $\text{E}$ and $\text{P}$, an ordinarily reflected lightray won't hit $\text{E}$ ever again. Can this method still be applied to measuring distance, further on velocity, in a spacetime manifold?