## Vector Decomposition

- Details
- Category: Math
- Published on Sunday, 30 September 2012 19:33
- Hits: 18715

Vector decomposition is an important method of analyzing forces in systems, such as with ropes in technical rescue. Related fields of study include mechanical statics and, more fundamentally, trigonometry.

# Basics

Scalars and vectors are the two types of physical quantities. Scalars have a value, whereas vectors have a value and a direction. The quantity "50 pounds" is a scalar, and "5 miles north of Guam" is a vector. Vectors can be added, subtracted, and, in a few different ways, multiplied. Navigation is one of the most common examples of vector addition. The direction "2 blocks north and 3 blocks west from your starting location" is a good example of vector addition in navigation.

# Trigonometry Foundation

A good list of trigonometric identities is available on Wikipedia. The majority of technical rescue loads calculations can be completed using only 3 basic identities related to a right triangle. In the triangle below, a, b, and c refer to the lengths of the sides of the triangle. A, B, and C refer to the angles of the corners. Being a right triangle, C is defined as 90 degrees.

\[ \sin (A) = \frac{opposite}{hypotenuse} = \frac{a}{c} \]

\[ \cos (A) = \frac{adjacent}{hypotenuse} = \frac{b}{c} \]

\[ \tan (A) = \frac{opposite}{adjacent} = \frac{a}{b} \]

These are the definitions for the Sine, Cosine and Tangent functions. The important, and useful, feature of these functions are that they relate the relative lengths of the sides of a right triangle to it's angles.

An additional feature of triangles is that the sum of the angles of the 3 corners is always 180 degrees. Since we will be using right triangles exclusively, the angle of the corner labeled C is always 90 degrees. This means that if we know the angle of either A or B, we can find the angles of all the corners by solving the equation

\[ A + B + C = 180 degrees \]

# Components of a Vector

It can be extremely useful to break a vector up into components which are parallel to our system of axes. Taking the image below as an example, we would like to show what two vectors, parallel to the x and y axes, will add up to equal the same resultant of the **F** vector. We will call these two components **F**_{x} and **F**_{y}.

Looking at this in terms of a right triangle, we can see that it is analogous to the triangle in the previous section. Replacing the appropriate variables, we can see that

\[ \cos (\theta) = \frac{opposite}{hypotenuse} = \frac{\textbf{F}_x}{\textbf{F}} \]

and

\ \sin (\theta) = \frac{opposite}{hypotenuse} = \frac{\textbf{F}_y}{\textbf{F}}. \]

Solving for Fx and Fy,

\[ \textbf{F}_x = \textbf{F} \cos(\theta) \]

\[ \textbf{F}_y = \textbf{F} \sin(\theta). \]