MCH T 111 Handout - Resultants by Successive Triangles Consider the coplanar vectors shown in the figure below. We need to develop a procedure for determining the magnitude and the direction of the resultant of the force system using force triangles.

B

C

A

θ1 θ2 θ3

Coplanar Vector System

1. Proportionally sketch the first two vectors to be added. The tail of the second vector is placed at the tip of the first vector.

θ1 A

B

θ2 θ1

4. Keep adding in this fashion for additional vectors. Report the answer being sure to include units!

C θ3

γ θ

R

θ R

2 cos

sin sin

sinC sin

Law of Cosines

Law of Sines

The direction of R from the negative x-axis is:

θ = 180° − β - γ

R1

β

β

R1

θ1 A

B

θ2 θ1

α

2 cos

sin sin

sinB sin

Law of Cosines

Law of Sines

The direction of R1 from the positive x-axis is:

β = α + θ1

The first two vectors drawn in “tip-to-tail” fashion are shown here. Any two of the vectors could have been chosen as the first two.

origin

end

2. The resultant of the first pair of vectors is sketched from the origin of the first vector to the end of the second vector. Direction is absolutely important. Use trigonometry to determine the magnitude and direction of the first resultant.

origin

end

3. The third vector is then added to the resultant of the first two vectors in the same fashion. Sketch the first resultant. At the end of the first resultant, sketch the third force. The final resultant is directed from the origin of the first vector to the end of the third vector. Calculate the magnitude and direction of the final resultant using trigonometry.