The problem is to find the resultant force ($R$) and its direction with respect to the positive $x$-axis, given three forces:

1. $F_1 = 15\sqrt{2} \, \text{N}$, directed at a 45° angle from 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.

Solution:

Step 1: Resolve $F_1$ into $x$- and $y$-components.

Given that $F_1 = 15\sqrt{2} \, \text{N}$ is at a 45° angle:

• $F_{1x} = F_1 \cos(45^\circ) = 15\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 15 \, \text{N}$
• $F_{1y} = F_1 \sin(45^\circ) = 15\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 15 \, \text{N}$

Step 2: Identify the $x$- and $y$-components of $F_2$ and $F_3$.

• $F_2$ acts along the negative $x$-axis, so:
  • $F_{2x} = -10 \, \text{N}$
  • $F_{2y} = 0 \, \text{N}$
• $F_3$ acts along the negative $y$-axis, so:
  • $F_{3x} = 0 \, \text{N}$
  • $F_{3y} = -20 \, \text{N}$

Step 3: Calculate the total $x$- and $y$-components.

$R_x = F_{1x} + F_{2x} + F_{3x} = 15 - 10 + 0 = 5 \, \text{N}$ $R_y = F_{1y} + F_{2y} + F_{3y} = 15 + 0 - 20 = -5 \, \text{N}$

Step 4: Calculate the magnitude of the resultant force $R$.

$R = \sqrt{R_x^2 + R_y^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \, \text{N}$

Step 5: Determine the angle $\theta$ of $R$ with respect to the positive $x$-axis.

$\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{-5}{5}\right) = \tan^{-1}(-1) = -45^\circ$ Since $R_y$ is negative and $R_x$ is positive, the resultant vector $R$ is directed 45° below the positive $x$-axis.

Final Answer:

• The magnitude of the resultant force $R$ is $5\sqrt{2} \, \text{N}$.
• The direction of $R$ is 45° below the positive $x$-axis (or $-45^\circ$ with respect to the positive $x$-axis).

Would you like a breakdown of any step or additional clarification?

1. How can we verify the resultant direction using vector addition?
2. What would change if the angle of $F_1$ were different?
3. How does knowing the direction help in practical applications?
4. What is the impact of each force component on the resultant?
5. How could this problem change if we used different coordinate systems?

Tip: Always resolve forces into components to simplify calculations in vector problems.