# Vector Addition

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}= [F_{1}^{2}+ F_{2}^{2}− 2 F_{1}F_{2}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 - Adding Forces

A force 1 with magnitude *3 kN* is acting in direction *80*^{o }from a force 2 with 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}

### Example - Airplane in Wind

A headwind of *100 km/h* is acting *30 ^{o} *starboard on an airplane with velocity

*900 km/h*.

The resulting velocity for the airplane related to the ground can be calculated as

v_{R}= [(900 km/h)^{2}+ (100 km/h)^{2}- 2 (900 km/h) (100 km/h) cos(180^{o}- (30^{o}))]^{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((180^{o}) - (30^{o})) / (815 km/h)]*

* = 3.5 ^{o}*

### Vector Calculator

The generic calculator below is based on equation *(1)* and can be used to add vectors quantities like velocities, forces etc.

### Parallelogram

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

- draw the vectors with right direction and magnitude
- draw parallel lines to the vectors
- draw the resultant vector to the crossing point between the parallel lines
- 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.

- draw the resultant vectors between two and two vectors
- draw the resultant vectors between two and two of resultant vectors
- continue until there is only one final resultant vector
- 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 F_{1} and F_{2}, and the resultant F_{(3,4)} by adding F_{3} and F_{4}. The find the resultant F_{(1,2.3,4)} by adding F_{(1,2)} and F_{(3,4)}.

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### • Statics

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