Vector Decomposition

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.

Trig Identities Triangle

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

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

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

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 Fx and Fy.

Vectors

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,

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

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

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