In this Wikipedia article https://en.m.wikipedia.org/wiki/Proper_reference_frame_(flat_spacetime) there was a confusing equation that involved the product of a bivector with a vector: $\frac{de_{(n)}}{d\tau}=-((U\wedge A)e_{(n)} + R\cdot e_{(n)}$)
$U$ is $4$-velocity, $A$ is $4$-acceleration, $\tau$ is proper time, and $R$ is the Riemann curvature tensor.
My question is what the dot product between a vector and tensor is along with what the product between a bivector and a vector is.
In ch. 6 of [1] the authors use the notation \begin{align} \frac{d\boldsymbol{e}_{\alpha'}}{d\tau}&=\boldsymbol{u}(\boldsymbol{a}\cdot \boldsymbol{e}_{\alpha'})-\boldsymbol{a}(\boldsymbol{u}\cdot\boldsymbol{e}_{\alpha'})\,. \end{align} Because $\boldsymbol{e}_{\alpha'},\boldsymbol{u},\boldsymbol{a}$ are vectors this can be written in index notation: \begin{align} \frac{d{e_{\alpha'}}^\mu}{d\tau}&=u^\mu a^\nu e_{\alpha'\nu}-a^\mu u^\nu e_{\alpha'\nu}\,. \end{align} The link you provided also has some index notation which I find a bit confusing because of the $\mu$ they are using twice: \begin{align} \frac{d\boldsymbol{e}_{(\mu)}}{d\tau}&=-\boldsymbol{\vartheta}\boldsymbol{e}_{(\mu)}\,,\\ \vartheta^{\mu\nu}&=A^\mu U^\nu-A^\nu U^\mu+U_\alpha\omega_\beta e^{\alpha\beta\mu\nu}\,. \end{align} I would rather write this as \begin{align} \frac{d\boldsymbol{e}_{(\alpha)}}{d\tau}&=-\boldsymbol{\vartheta}\boldsymbol{e}_{(\alpha)}\, \end{align} so that for each $\alpha=0,1,2,3\,,$ $$ \frac{d{e_{(\alpha)}}^\mu}{d\tau}=-\vartheta^{\mu\nu}\,{e_{(\alpha)\nu}}\,. $$ The dot product of the vector $\boldsymbol{e}_{(\alpha)}$ with those various parts of $\boldsymbol{\vartheta}$ is just the normal contraction of a second rank tensor with the components of a vector.
[1] Misner, Thorne & Wheeler, Gravitation.