1. Volume of a Tetrahedron:
- The volume $V$ of a tetrahedron with coterminous sides $\bar{a}, \bar{b}, \bar{c}$ is given by:
$$
V=\frac{1}{6}|\bar{a} \cdot(\bar{b} \times \bar{c})|
$$
- Given that the volume $V=3$, we have:
$$
\begin{aligned}
& \frac{1}{6}|\bar{a} \cdot(\bar{b} \times \bar{c})|=3 \\
& |\bar{a} \cdot(\bar{b} \times \bar{c})|=18
\end{aligned}
$$
2. Vector Triple Product Expression:
- The given expression is a known identity related to vector calculus:
$$
(\bar{r} \cdot \bar{a})(\bar{b} \times \bar{c})+(\bar{r} \cdot \bar{b})(\bar{c} \times \bar{a})+(\bar{r} \cdot \bar{c})(\bar{a} \times \bar{b})=\bar{r} \cdot(\bar{a} \times(\bar{b} \times \bar{c}))
$$
3. Magnitude of $\bar{r}$ :
- Given $|\bar{r}|=2$, the expression then becomes proportional to $|\bar{a} \cdot(\bar{b} \times \bar{c})|$, which is 18 .
Solving the Expression:
Since $|\bar{a} \cdot(\bar{b} \times \bar{c})|=18$, and knowing the form of the expanded expression, we substitute into the simplified form:
- The expression $\bar{r} \cdot(\bar{a} \times(\bar{b} \times \bar{c}))$ essentially reflects the same scalar triple product scaled by $|\bar{r}|$, which is 2 .
Thus, the absolute value of the given expression is:
$$
|2 \times 18|=36
$$
Therefore, the value of the expression is 36 .
