mathematical solution

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

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.

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[ F_1 · sin(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 3 kN is acting in a direction 80^o from 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