Charmed hadron spectroscopy is a quickly progressing field that has enjoyed a growing interest in recent years from both theorist and experimentalists. Open-charmed hadrons consist of a heavy $c$ quark bound to a light quark ($u$, $d$, $s$ quarks). These particles are in particular spin-parity states making for a very interesting quantum mechanical system allowing the study of the QCD potential holding them together. Decays of type $B\rightarrow D^{}DK$ offer a rich environment to study charmed spectroscopy thanks to their abundance and low background. Using $pp$ collision data at 7, 8, and 13 TeV center-of-mass energies corresponding to an integrated luminosity of $9.1\mathrm{fb}^{-1}$ collected with the LHCb detector at CERN's Large Hadron Collider, this thesis presents the most accurate measurement to-date of four branching fraction ratios of $B\rightarrow D^{}DK$ modes. These are measured to be \begin{equation*} \begin{split} \frac{\mathcal{B} (B^+ \rightarrow D^{+}D^-K^+)}{\mathcal{B} (B^+ \rightarrow \overline{D}^{0}D^0K^+)} &= 0.590 \pm 0.017 (\mathrm{stat}) \pm 0.008 (\mathrm{syst}) \pm 0.013 (\mathcal{B}(D)) , \[10pt] \frac{\mathcal{B} (B^+ \rightarrow D^{-}D^+K^+)}{\mathcal{B} (B^+ \rightarrow \overline{D}^{0}D^0K^+)} &= 0.524 \pm 0.015 (\mathrm{stat}) \pm 0.007 (\mathrm{syst}) \pm 0.012 (\mathcal{B}(D)) , \[10pt] \frac{\mathcal{B} (B^0 \rightarrow D^{-}D^0K^+)}{\mathcal{B} (B^0 \rightarrow D^{-}D^0K^+} &= 1.767 \pm 0.028 (\mathrm{stat}) \pm 0.018 (\mathrm{syst}) \pm 0.036 (\mathcal{B}(D)) , \[10pt] \frac{\mathcal{B} (B^+ \rightarrow D^{+}D^-K^+)}{\mathcal{B} (B^+ \rightarrow D^{-}D^+K^+)} &= 1.098 \pm 0.040 (\mathrm{stat}) \pm 0.017 (\mathrm{syst}), \end{split} \end{equation} where the first uncertainty is statistical, the second systematic, and the third is due to the uncertainties on the $D$ meson decay branching fractions measured by other experiments. Furthermore, this thesis shows the Dalitz plots of the final states for these decay modes using data samples an order of magnitude larger than samples collected by other experiments. These measurements are an important step towards a full amplitude analysis of $B\rightarrow D^{*}DK$ decays.

In addition, this thesis summarizes the installation and studies of the BCAM system used to monitor the position of the Inner Tracker detector of LHCb.