For part a), I know that $\int(F\cdot ds)$ = $\int F(c(t)\cdot c'(t)) dt$.
So for my first path, I use points A and B to get a vector:
$$v = \langle 1-0,0-0,2-0 \rangle = \langle 1,0,2\rangle$$
Then, I get the following parametric equations:
$x = t, y =0, z = 2t$.
I can use this as my $c(t)$, but how do I find the bounds of my integral?
On
$$\int_\mathbf\sigma\mathbf F\cdot d\mathbf r = \int_{\mathbf\sigma_1}\mathbf F\cdot d\mathbf r +\int_{\mathbf\sigma_2}\mathbf F\cdot d\mathbf r$$
Where $\mathbf\sigma_1$ is the segment from $A$ to $B$, and $\mathbf\sigma_2$ is the segment from $B$ to $C$. Using dummy parameterizations, $$\mathbf\sigma_1 (t) = \langle t,0,2t\rangle \\ \mathbf\sigma_2 (t) = \langle 0,2t,-2t\rangle$$ In both cases, the velocity vectors $\mathbf\sigma_1' (t)$ and $\mathbf\sigma_2' (t)$ are constant. And clearly, $0 \leq t\leq 1$
Hint:
You can write the parametric equation of the segment between $A=(x_A,y_A,z_A)$ and $B=(x_B,y_B,z_B)$ as: $$ (x,y,z)=(x_A,y_A,z_A)+t(x_A-x_B,y_A-y_B,z_A-z_B) $$ with $t \in [0,1]$