Let's solve this step-by-step:

Given:

• $F_1 = 10 \, \text{N}$ at an angle of $45^\circ$ from the positive $Y$-axis.
• $F_2 = 8 \, \text{N}$ at an angle of $30^\circ$ from the negative $X$-axis.

We need to find:

1. The magnitude of the resultant vector ($R$).
2. The direction of the resultant vector ($\theta$ from a specified axis).
3. A diagram of the resultant vector.

Step 1: Resolve $F_1$ and $F_2$ into their components

For $F_1$:

• $F_{1x} = F_1 \cdot \sin(45^\circ) = 10 \cdot \sin(45^\circ) = 10 \cdot \frac{\sqrt{2}}{2} = 7.07 \, \text{N}$ (positive along the X-axis)
• $F_{1y} = F_1 \cdot \cos(45^\circ) = 10 \cdot \cos(45^\circ) = 10 \cdot \frac{\sqrt{2}}{2} = 7.07 \, \text{N}$ (positive along the Y-axis)

For $F_2$:

• $F_{2x} = F_2 \cdot \cos(30^\circ) = 8 \cdot \cos(30^\circ) = 8 \cdot \frac{\sqrt{3}}{2} = 6.93 \, \text{N}$ (negative along the X-axis)
• $F_{2y} = F_2 \cdot \sin(30^\circ) = 8 \cdot \sin(30^\circ) = 8 \cdot \frac{1}{2} = 4 \, \text{N}$ (negative along the Y-axis)

Step 2: Find the resultant components $R_x$ and $R_y$

• $R_x = F_{1x} + F_{2x} = 7.07 - 6.93 = 0.14 \, \text{N}$
• $R_y = F_{1y} + F_{2y} = 7.07 - 4 = 3.07 \, \text{N}$

Step 3: Calculate the magnitude of the resultant vector $R$

$R = \sqrt{(R_x)^2 + (R_y)^2} = \sqrt{(0.14)^2 + (3.07)^2} = \sqrt{0.0196 + 9.4249} = \sqrt{9.4445} \approx 3.07 \, \text{N}$

Step 4: Determine the direction of the resultant vector $\theta$

$\theta = \arctan\left(\frac{R_y}{R_x}\right) = \arctan\left(\frac{3.07}{0.14}\right) \approx 87.39^\circ$

This angle is measured from the positive $X$-axis towards the $Y$-axis.

Answers:

1. Magnitude of the resultant vector: $R \approx 3.07 \, \text{N}$
2. Direction of the resultant vector: $\theta \approx 87.39^\circ$ from the positive $X$-axis.
3. Diagram: I can describe it, or you can draw it based on the components $R_x$ and $R_y$ with the resultant angle.

Would you like further details on any of these steps or assistance with the diagram?

Here are some related questions to explore further:

1. How would the result change if the angles were measured from a different reference axis?
2. What would happen to the resultant vector if $F_1$ and $F_2$ had the same magnitude?
3. How does changing the angle of $F_1$ affect the magnitude and direction of the resultant vector?
4. What is the physical interpretation of vector addition in terms of force equilibrium?
5. How can we use vector components to simplify complex force systems in physics?

Tip: When working with vectors, resolving into components and using trigonometric identities simplifies calculations greatly.