ADDITION OF VECTORS

• ANALYTICAL METHOD– Vector addition by components1. Resolve the vectors into their components in the

x and y direction. Ax = A cos θ, Ay = A sin θ

ADDITION OF VECTORS

2. Add the components in the x direction to give Rx and add the components in the y direction to give ry. That is

Rx = Ax + Bx + Cx = sum of x-componentsRy = Ay + By + Cy = sum of y-components

ADDITION OF VECTORS

3. Find the magnitude of the resultant R from the components Rx and Ry. From the Pythagorean Theorem

R = √(Rx)² + (Ry)²

ADDITION OF VECTORS

4. Determine the direction of the resultant R using Tan θ = Ry

Rx

θ = inv tan Ry Rx

SAMPLE PROBLEM

• GIVEN: A = 2 cm, N; B = 3 cm, 20° N of EC = 5 cm, 40° S or E; D = 4 cm, 40° N or W

Ax = A cos θ = (2cm) (cos 90°) = 0Ay = A sin θ = (2cm) (sine 90°) = 2cmBx = B cos θ = (3cm) (cos 20°) = 2.82.cmBy = B sin θ = (3cm) (sin 20°) = 1.03 cm

SAMPLE PROBLEM

Cx = C cos θ = (5cm) (cos 40°) = 3.83 cmCy = C sin θ = (5cm) (sin 40°) = -3.21 cmDx = D cos θ = (4cm) (cos 40°) = -3.06 cmDy = D sin θ = (4cm) (sin 40°) = 2.57 cmRx = Ax + Bx + Cx + Dx = 0 + 2.82 +3.83 + (-3.06) = 3.59 cmRy = Ay + By + Cy + Dy

= 2 + 1.03 + (-3.21) +2.57 = 2.39 cm

SAMPLE PROBLEM

R = √(Rx)² + (Ry) ² = √(3.59 cm) ² + (2.39 cm) ²R = 4.31 cm

Tan θ = Ry / Rx = 2.39 cm / 3.59 cm = 0.6657 θ = inv tan 0.6657 = 33° 39’ 11”R = 4.31 cm, 33° 39’ 11”