Express the Cartesian components of force F1.

The magnitude of $\vec{F_1}$ is $30 \, \text{N}$ at an angle of $45^\circ$ from the positive x-axis.

$$F_{1x} = F_1 \cos(45^\circ) = 30 \, \text{N} \times \frac{\sqrt{2}}{2} \approx 21.213 \, \text{N}$$ $$F_{1y} = F_1 \sin(45^\circ) = 30 \, \text{N} \times \frac{\sqrt{2}}{2} \approx 21.213 \, \text{N}$$

The magnitude of $\vec{F_2}$ is $40 \, \text{N}$ at an angle of $15^\circ$ below the positive x-axis, which is $345^\circ$ from the positive x-axis.

$$F_{2x} = F_2 \cos(345^\circ) = 40 \, \text{N} \times \cos(15^\circ) \approx 40 \, \text{N} \times 0.9659 \approx 38.636 \, \text{N}$$ $$F_{2y} = F_2 \sin(345^\circ) = 40 \, \text{N} \times (-\sin(15^\circ)) \approx 40 \, \text{N} \times (-0.2588) \approx -10.352 \, \text{N}$$

The magnitude of $\vec{F_3}$ is $25 \, \text{N}$ at an angle of $15^\circ$ from the negative y-axis towards the negative x-axis. This corresponds to an angle of $270^\circ - 15^\circ = 255^\circ$ from the positive x-axis.

$$F_{3x} = F_3 \cos(255^\circ) = 25 \, \text{N} \times (-\sin(15^\circ)) \approx 25 \, \text{N} \times (-0.2588) \approx -6.470 \, \text{N}$$ $$F_{3y} = F_3 \sin(255^\circ) = 25 \, \text{N} \times (-\cos(15^\circ)) \approx 25 \, \text{N} \times (-0.9659) \approx -24.148 \, \text{N}$$

The Cartesian components for each force are:

• $\vec{F_1} = \boxed{(21.213 \, \text{N})\hat{i} + (21.213 \, \text{N})\hat{j}}$
• $\vec{F_2} = \boxed{(38.636 \, \text{N})\hat{i} - (10.352 \, \text{N})\hat{j}}$
• $\vec{F_3} = \boxed{(-6.470 \, \text{N})\hat{i} - (24.148 \, \text{N})\hat{j}}$