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.

F_R = [ F₁² + F₂² − 2 F₁ F₂ cos(180^o - (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 [ F₁sin(180^o - (α + β)) / F_R] (2)

where

α + β = the angle between vector 1 and 2 is known

Example - Calculating Vector Forces

A force 1 of magnitude 5 kN is acting in a direction 80^ofrom a force 2 of magnitude 8 kN.

The resulting force can be calculated as

F_R = [ (3(kN))² + (8(kN))² - 2 5(kN) 8(kN) cos(180^o - (80^o)) ]^1/2

= 9 kN

The angle between vector 1 and the resulting vector can be calculated as

α = sin^-1[ 3(kN) sin(180^o - (80^o)) / 9(kN)]

= 19.1^o

The angle between vector 2 and the resulting vector can be calculated as

α = sin^-1[ 8(kN) sin(180^o - (80^o)) / 9(kN)]

= 60.9^o

Online Vector Calculator

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