Calculate the resultant vector using: A. Polygon Method B. Component Method Given: F₁ = 55 N, 33° E of N F₂ = 30 N, 40° W of S F₃ = 80 N, 10° S of E

Steps to Solve

1. Convert the Angles to Standard Position

   For each force, convert the angles to standard position (measured from the positive x-axis, counter-clockwise):

   • For ( F_1 = 55 , N, , 33^\circ , E , of , N ):
     • Angle = ( 90^\circ - 33^\circ = 57^\circ )
   • For ( F_2 = 30 , N, , 40^\circ , W , of , S ):
     • Angle = ( 270^\circ + 40^\circ = 310^\circ )
   • For ( F_3 = 80 , N, , 10^\circ , S , of , E ):
     • Angle = ( 0^\circ + 10^\circ = 10^\circ )

2. Calculate Components for Each Force

   Use the formulas ( F_x = F \cdot \cos(\theta) ) and ( F_y = F \cdot \sin(\theta) ).

   • For ( F_1 ):
     • ( F_{1x} = 55 \cdot \cos(57^\circ) )
     • ( F_{1y} = 55 \cdot \sin(57^\circ) )
   • For ( F_2 ):
     • ( F_{2x} = 30 \cdot \cos(310^\circ) )
     • ( F_{2y} = 30 \cdot \sin(310^\circ) )
   • For ( F_3 ):
     • ( F_{3x} = 80 \cdot \cos(10^\circ) )
     • ( F_{3y} = 80 \cdot \sin(10^\circ) )

3. Calculate Resultant Components

   Sum the components in the x and y directions:

   [ R_x = F_{1x} + F_{2x} + F_{3x} ] [ R_y = F_{1y} + F_{2y} + F_{3y} ]

4. Calculate the Magnitude and Direction of the Resultant Vector

   Use the Pythagorean theorem to find the magnitude:

   [ R = \sqrt{R_x^2 + R_y^2} ]

   Find the direction (angle) using:

   [ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) ]

5. Graphical Representation (Polygon Method)

   • Draw each force vector to scale based on the calculations.
   • Begin with ( F_1 ), then place ( F_2 ) from the tip of ( F_1 ), and ( F_3 ) from the tip of ( F_2 ).
   • The resultant vector ( R ) can be drawn from the tail of ( F_1 ) to the tip of ( F_3 ).