To solve this problem, we calculate the line integral of the vector field $\mathbf{F}$ along the path $C$, which consists of two line segments. Let's denote the segments as $C_1$ (from $(0, 0, 2)$ to $(3, 0, 0)$) and $C_2$ (from $(3, 0, 0)$ to $(0, 2, 1)$).

The line integral is computed as: $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_{C_1} \mathbf{F} \cdot d\mathbf{r} + \int_{C_2} \mathbf{F} \cdot d\mathbf{r}.$

Step 1: Parametrize $C_1$

The line segment $C_1$ can be parametrized as: $\mathbf{r}_1(t) = (1-t)(0, 0, 2) + t(3, 0, 0) = (3t, 0, 2-2t), \quad t \in [0, 1].$ The derivative is: $\frac{d\mathbf{r}_1}{dt} = (3, 0, -2).$

Step 2: Parametrize $C_2$

The line segment $C_2$ can be parametrized as: $\mathbf{r}_2(t) = (1-t)(3, 0, 0) + t(0, 2, 1) = (3-3t, 2t, t), \quad t \in [0, 1].$ The derivative is: $\frac{d\mathbf{r}_2}{dt} = (-3, 2, 1).$

Step 3: Compute the line integrals

To compute each integral, we need the expression for $\mathbf{F}(\mathbf{r})$. If $\mathbf{F}$ is given, we substitute it into: $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_C \mathbf{F} \cdot \frac{d\mathbf{r}}{dt} \, dt.$ If the vector field is unknown, we need it to proceed. Could you provide $\mathbf{F}$?