To solve this problem, we’ll break each force into its horizontal ($x$) and vertical ($y$) components, sum these components to find the resultant vector, and calculate the magnitude and direction of the resultant force.

Step 1: Break down the forces into components

1. Force 1: 120 pounds at $70^\circ$ south of east ($S70^\circ E$)

   • Angle measured clockwise from the positive $x$-axis: $360^\circ - 70^\circ = 290^\circ$.
   • $x_1 = 120 \cos(290^\circ)$
   • $y_1 = 120 \sin(290^\circ)$

2. Force 2: 80 pounds at $54^\circ$ north of east ($N54^\circ E$)

   • Angle measured counterclockwise from the positive $x$-axis: $54^\circ$.
   • $x_2 = 80 \cos(54^\circ)$
   • $y_2 = 80 \sin(54^\circ)$

Step 2: Sum the components

• Total $x$-component: $x_R = x_1 + x_2$
• Total $y$-component: $y_R = y_1 + y_2$

Step 3: Calculate the magnitude and direction of the resultant

• Magnitude: $R = \sqrt{x_R^2 + y_R^2}$
• Direction: $\theta = \tan^{-1}\left(\frac{y_R}{x_R}\right)$

Let me calculate this for you.import math

Components of Force 1 (120 pounds, S70°E)

force1_magnitude = 120 force1_angle_deg = 290 # S70°E = 360° - 70° force1_angle_rad = math.radians(force1_angle_deg) x1 = force1_magnitude * math.cos(force1_angle_rad) y1 = force1_magnitude * math.sin(force1_angle_rad)

Components of Force 2 (80 pounds, N54°E)

force2_magnitude = 80 force2_angle_deg = 54 # N54°E force2_angle_rad = math.radians(force2_angle_deg) x2 = force2_magnitude * math.cos(force2_angle_rad) y2 = force2_magnitude * math.sin(force2_angle_rad)

Resultant components

x_R = x1 + x2 y_R = y1 + y2

Magnitude of the resultant

R = math.sqrt(x_R2 + y_R2)

Direction of the resultant (angle with respect to the positive x-axis)

theta_rad = math.atan2(y_R, x_R) theta_deg = math.degrees(theta_rad)

R, theta_deg(100.31697888872883, -28.613553811860676)(100.31697888872883, -28.613553811860676)successThe magnitude of the resultant force is approximately 100.32 pounds, and the direction is approximately -28.6° (28.6° below the positive $x$-axis).

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Related Questions

1. How do we determine whether angles are measured clockwise or counterclockwise?
2. Why are trigonometric functions used to decompose vectors into components?
3. What would change if the forces acted at different angles?
4. How does the resultant direction depend on the sign of the components?
5. Can this method be applied to forces acting in three dimensions?

Tip

Always double-check the quadrant of the resultant vector when interpreting the angle to ensure its direction is correct.