Applied Mechanics and Graphic Statics
The resolved part of the resultant of two forces inclined at an angle θ in a given direction is

Arithmetical sum of the resolved parts of the forces in the direction
Algebraic sum of the resolved parts of the forces in the direction
Difference of the forces multiplied by cosine θ°
Sum of the forces multiplied by the tangent θ°

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Applied Mechanics and Graphic Statics
A body of weight ‘w’ placed on an inclined plane is acted upon by a force ‘P’ parallel to the plane which causes the body just to move up the plane. If the angle of inclination of the plane is ‘θ’ and angle of friction is ‘φ’, the minimum value of ‘P’, is

w cos (θ + φ)/cos φ
w sin (θ - φ)/cos φ
w sin (φ - θ)/cos φ
w sinθ cos(θ - φ)/sin φ

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