Vector Addition
Online vector calculator - add vectors with different magnitude and direction - like forces, velocities and more.
In mechanics there are two kind of quantities
- scalar quantities with magnitude - time, temperature, mass etc.
- vector quantities with magnitude and direction - velocity, force etc.
When adding vector quantities both magnitude and direction are important. Common methods adding coplanar vectors (vectors acting in the same plane) are
- the parallelogram law
- the triangle rule
- trigonometric calculation
The Parallelogram Law
The procedure of "the parallelogram of vectors addition method" is
- draw vector 1 using appropriate scale and in the direction of its action
- from the tail of vector 1 draw vector 2 using the same scale in the direction of its action
- complete the parallelogram by using vector 1 and 2 as sides of the parallelogram
- the resulting vector is represented in both magnitude and direction by the diagonal of the parallelogram
The Triangle Rule
The procedure of "the triangle of vectors addition method" is
- draw vector 1 using appropriate scale and in the direction of its action
- from the nose of the vector draw vector 2 using the same scale and in the direction of its action
- the resulting vector is represented in both magnitude and direction by the vector drawn from the tail of vector 1 to the nose of vector 2
Trigonometric Calculation
The resulting vector of two coplanar vector can be calculated by trigonometry using "the cosine rule" for a non-right-angled triangle.
FR = [F12 + F22 − 2 F1 F2 cos(180o - (α + β))]1/2 (1)
where
F = the vector quantity - force, velocity etc.
α + β = angle between vector 1 and 2
The angle between the vector and the resulting vector can be calculated using "the sine rule" for a non-right-angled triangle.
α = sin-1[F1 sin(180o - (α + β)) / FR] (2)
where
α + β = the angle between vector 1 and 2 is known
Example - Adding Forces
A force 1 with magnitude 3 kN is acting in direction 80o from a force 2 with magnitude 8 kN.
The resulting force can be calculated as
FR = [(3 kN)2 + (8 kN)2 - 2 (5 kN) (8 kN) cos(180o - (80o))]1/2
= 9 (kN)
The angle between vector 1 and the resulting vector can be calculated as
α = sin-1[ (3 kN) sin(180o - (80o)) / (9 kN) ]
= 19.1o
The angle between vector 2 and the resulting vector can be calculated as
α = sin-1[ (8 kN) sin(180o - (80o)) / (9 kN) ]
= 60.9o
Example - Airplane in Wind
A headwind of 100 km/h is acting 30o starboard on an airplane with velocity 900 km/h.
The resulting velocity for the airplane related to the ground can be calculated as
vR = [(900 km/h)2 + (100 km/h)2 - 2 (900 km/h) (100 km/h) cos(180o - (30o))]1/2
= 815 (km/h)
The angle between the airplane course and actual relative ground course can be calculated as
α = sin-1[ (100 km/h) sin((180o) - (30o)) / (815 km/h) ]
= 3.5o
Vector Calculator
The generic calculator below is based on equation (1) and can be used to add vectors quantities like velocities, forces etc.
magnitude of vector 1 - F1
magnitude of vector 2 - F2
angle between vector 1 and 2 (degrees)
Parallelogram
Resultant vectors can be estimated by drawing parallelograms as indicated below.
- draw the vectors with right direction and magnitude
- draw parallel lines to the vectors
- draw the resultant vector to the crossing point between the parallel lines
- measure the magnitude and direction of the resultant vector in the drawing
The method can also be used with more than two vectors as indicated below.
- draw the resultant vectors between two and two vectors
- draw the resultant vectors between two and two of resultant vectors
- continue until there is only one final resultant vector
- measure direction and magnitude of the final resultant vector in the drawing
In the example above - first find the resultant F(1,2) by adding F1 and F2, and the resultant F(3,4) by adding F3 and F4. The find the resultant F(1,2.3,4) by adding F(1,2) and F(3,4).