Vector Addition

Online vector calculator - add vectors with different magnitude and direction

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

vector addition 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

vector addition 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

vector addition cosine rule

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[ Fsin(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

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 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 airplane course and actual ground course can be calculated as

 α = sin-1[ (100 km/h) sin(180o - (30o)) / (815 km/h) ]

    = 3.5o

airplane wind velocity

Online Vector Calculator

The generic calculator below can used to add vectors for velocity, forces etc.

 Magnitude of vector 1 - F1

Magnitude of vector 2 - F2

angle between vector 1 and 2 (degrees)

Related Topics

  • Basics - Basic Information as SI-system, Unit converters, Physical constants
  • Dynamics - Dynamics Motion - velocity and acceleration
  • Mechanics - Kinematics, forces, vectors, motion, momentum, energy and the dynamics of objects
  • Mathematics - Mathematical rules and laws - areas, volumes, exponents, trigonometric functions and more
  • Statics - Loads - force and torque

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