Vector Addition
Online vector calculator - add vectors with different magnitude and direction
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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 - Calculating Vector Forces
A force 1 of magnitude 3 kN is acting in a direction 80o from a force 2 of 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
Online Vector Calculator
The generic calculator below can used to add vectors for velocity, forces etc.
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Related Topics
- Basics - Basic Information as SI-system, Unit converters, Physical constants...
- Dynamics - Dynamics Motion - velocity and acceleration
- Mathematics - Mathematical rules and laws - areas, volumes, exponents, trigometric functions and more
- Mechanics - Kinematics, forces, vectors, motion, momentum, energy and the dynamics of objects
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