# Addition and Subtraction of Vectors

## Introduction

Some quantities in nature are completely described by a number. A scalar is a quantity that has only magnitude; in other words, a scalar is a quantity that requires only a number to be completely determined. Examples fo scalars are length, mass, time, temperature, work, energy, etc. Being simple numbers, scalars are added in the same way that we add real numbers.

There are some other quantities, those that possess both magnitude and direction. Thus, more than just a number is needed to completely define them. These quantities are called vectors. Then, a vector is a quantity that is more complex than a scalar since it has two characteristics: magnitude and direction. Examples of vector quantities are displacement, velocity, acceleration, force, impulse, etc. Graphically, vector quantities can be represented by a directed line segment (an arrow) drawn to scale. The length of the arrow is proportional to the magnitude of the vector, and the direction of the arrow is proportional to the magnitude of the vector, and the direction of the arrow represents the direction of the vector quantity.
Analytically, vectors are represented by a letter with an arrow on top of it. The magnitude of the vector is written by placing the letter with the arrow between absolute value bars.

In order to handle physical problems involving vectors, a mathematical theory known as vector algebra has been developed. We are going to state the laws and defing the operations of this mathematical theory, which is a very important tool for the physicist.

## Characteristics of a Vector

As stated before, a scalar is a quantity that needs only a number to be completely determined. A vector, however, is a mathematical expression that has both, magnitude and direction. Graphically, it is represented by an oriented segment. The length of the segment represents the magnitude of the vector. The point marked O (origin) is where the vector begins, and the point marked P is the extreme or end of the vector.

Analytically, a vector is represented by a letter with an arrow on top of it.
->
A
and its magnitude is indicated by the vector symbol enclosed in absolute value bars
-->
| A |
The magnitude of a vector is related to its size. It is a scalar which represents how big or how small is a vector with respect to a reference. The direction of a vector is the place where the vector is pointing; it is indicated by the head of the arrow.

## Useful Definitions

As we see, in physics it is not enough to deal with scalar quantities only, since a lot of physical quantities require a direction to be completely determined. We will state now the rules of the mathematical formalism that let us deal with vector quantities as if they are scalars. The following definitions will be useful for this task.

(a) Equal Vectors: Two vectors are equal if they have the same magnitude, the same direction, and the same point of application.

(b) Opposite Vectors: Two vectors are opposite if they have the same magnitude and indicate in exactly opposite directions.

(c) Resultant: A resultant vector is a single vector that produces the same physical effect that would be produced by two or more individual vectors acting together. Analytically, the resultant of two vectors is a third vector such that
--> -> ->
C=A+B
Graphically, we can add two vectors by using techniques like the parallelogram method, which will be dicussed later in this page.

(d) Difference: The difference of two vectors is a third vector such that
--> -> ->
D=A-B
Notice that, algebraically speaking, this is the same as
-> -> ->
D=A+(-B)
so the concepts of opposite vector and resultant can be used to obtain graphically the diffrence of two vectors, which happens to be the resultant obtained from adding teh first vector with the opposite of the second vector.

(e) Unitary Vectors: Vectors with magnitude 1. The unitary vectors are normal vectors divided by their magnitudes:

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