To solve this problem, let's break down the forces given and their respective directions.

Given Data:

1. $F_1 = 15\sqrt{2} \, \text{N}$, directed at $45^\circ$ above the positive x-axis.
2. $F_2 = 10 \, \text{N}$, directed along the negative x-axis.
3. $F_3 = 20 \, \text{N}$, directed along the negative y-axis.

Step-by-Step Solution:

We need to resolve each force into its x and y components and then sum these components to find the resultant force and its angle relative to the positive x-axis.

1. Resolving $F_1$:

Since $F_1$ is at a $45^\circ$ angle to the positive x-axis: $F_{1x} = 15\sqrt{2} \cos(45^\circ) = 15 \, \text{N}$ $F_{1y} = 15\sqrt{2} \sin(45^\circ) = 15 \, \text{N}$

2. Resolving $F_2$:

Since $F_2$ is directed along the negative x-axis: $F_{2x} = -10 \, \text{N}$ $F_{2y} = 0 \, \text{N}$

3. Resolving $F_3$:

Since $F_3$ is directed along the negative y-axis: $F_{3x} = 0 \, \text{N}$ $F_{3y} = -20 \, \text{N}$

4. Summing the Components:

Now, we sum the x and y components separately.

Resultant x-component ($R_x$):

$R_x = F_{1x} + F_{2x} + F_{3x} = 15 - 10 + 0 = 5 \, \text{N}$

Resultant y-component ($R_y$):

$R_y = F_{1y} + F_{2y} + F_{3y} = 15 + 0 - 20 = -5 \, \text{N}$

5. Calculating the Resultant Force $R$:

The magnitude of the resultant force $R$ is given by: $R = \sqrt{R_x^2 + R_y^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \, \text{N}$

6. Finding the Angle $\theta$:

The angle $\theta$ relative to the positive x-axis is calculated using: $\theta = \tan^{-1} \left( \frac{R_y}{R_x} \right) = \tan^{-1} \left( \frac{-5}{5} \right) = \tan^{-1}(-1) = -45^\circ$

Since the angle is negative, the resultant force is directed $45^\circ$ below the positive x-axis.

Final Answer:

• Magnitude of the resultant force, $R$: $5\sqrt{2} \, \text{N}$
• Direction relative to the positive x-axis: $45^\circ$ below the positive x-axis.

Would you like more details on any step or further clarification?

Here are some related questions for further understanding:

1. How would the result change if $F_2$ were directed along the positive x-axis?
2. How can we find the direction if the resultant vector were in a different quadrant?
3. What is the physical interpretation of resolving forces into components?
4. How does the angle impact the effective force in the x and y directions?
5. Can you explain why the $45^\circ$ angle resulted in equal x and y components?

Tip: When breaking down forces, always use trigonometric identities for angles and be mindful of direction (positive or negative signs).