The final answer for some of these would require more information.
To get the full solution, better provide one problem at a time with full givens. The final answer for some of these would
The cable and pulley system is used to lift a weight $W$. Determine the tension $T$ in the cable. Draw a free-body diagram of the pulley system. 2: Write the equations of equilibrium Since the system is in equilibrium, we can write: $\sum F_x = 0$ $\sum F_y = 0$ 3: Solve for T Assuming the tension in the cable is $T$ and there are 3 pulleys, $W = 3T$ $T = \frac{W}{3}$ Determine the tension $T$ in the cable
$\mathbf{r}_{AB} = 0.2 \mathbf{i} + 0.1 \mathbf{j}$ $\mathbf{F} = 100 \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k}$ (Assuming F is along the x-axis) The final answer for some of these would
$\theta = \tan^{-1} \left( \frac{\mathbf{R}_y}{\mathbf{R}_x} \right) = \tan^{-1} \left( \frac{223.21}{186.60} \right) = 50.11^\circ$
The final answer is: $\boxed{\frac{W}{3}}$