Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications!

This is an AMP page - Open full page! for all features.

Vector Addition

Online vector calculator - add vectors with different magnitude and direction - like forces, velocities and more

Sponsored Links

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[Fsin(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)


Load Calculator!

Parallelogram

Resultant vectors can be estimated by drawing parallelograms as indicated below.

  1. draw the vectors with right direction and magnitude
  2. draw parallel lines to the vectors
  3. draw the resultant vector to the crossing point between the parallel lines
  4. 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.

  1. draw the resultant vectors between two and two vectors
  2. draw the resultant vectors between two and two of resultant vectors
  3. continue until there is only one final resultant vector
  4. 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).

Sponsored Links

Related Topics

Related Documents

Sponsored Links

Share

Search Engineering ToolBox

  • the most efficient way to navigate the Engineering ToolBox!

SketchUp Extension - Online 3D modeling!

Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with the amazing, fun and free SketchUp Make and SketchUp Pro .Add the Engineering ToolBox extension to your SketchUp from the Sketchup Extension Warehouse!

Privacy

We don't collect information from our users. Only emails and answers are saved in our archive. Cookies are only used in the browser to improve user experience.

Some of our calculators and applications let you save application data to your local computer. These applications will - due to browser restrictions - send data between your browser and our server. We don't save this data.

Google use cookies for serving our ads and handling visitor statistics. Please read Google Privacy & Terms for more information about how you can control adserving and the information collected.

AddThis use cookies for handling links to social media. Please read AddThis Privacy for more information.

Topics

Unit Converters

Temperature

oC
oF


Load Calculator!

Length

m
km
in
ft
yards
miles
naut miles


Load Calculator!

Area

m2
km2
in2
ft2
miles2
acres


Load Calculator!

Volume

m3
liters
in3
ft3
us gal


Load Calculator!

Weight

kgf
N
lbf


Load Calculator!

Velocity

m/s
km/h
ft/min
ft/s
mph
knots


Load Calculator!

Pressure

Pa (N/m2)
bar
mm H2O
kg/cm2
psi
inches H2O


Load Calculator!

Flow

m3/s
m3/h
US gpm
cfm


Load Calculator!

8 6

This website use cookies. By continuing to browse you are agreeing to our use of cookies! Learn more