# Vectors

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## Introduction

This chapter focuses on vectors. We will learn what is a vector and how it differs from everyday numbers. We will also learn how to add, subtract and multiply them and where they appear in Physics.

Are vectors Physics? No, vectors themselves are not Physics. Physics is just a description of the world around us. To describe something we need to use a language. The most common language used to describe Physics is Mathematics. Vectors form a very important part of the mathematical description of Physics, so much so that it is absolutely essential to master the use of vectors.

## Scalars and vectors

In Mathematics, you learned that a number is something that represents a quantity. For example if you have 5 books, 6 apples and 1 bicycle, the 5, 6, and 1 represent how many of each item you have.

These kinds of numbers are known as scalars .

Scalar

A scalar is a quantity that has only magnitude (size).

An extension to a scalar is a vector, which is a scalar with a direction. For example, if you travel 1 km down Main Road to school, the quantity 1 km down Main Road is a vector. The “ 1 km ” is the quantity (or scalar) and the “ down Main Road ” gives a direction.

In Physics we use the word magnitude to refer to the scalar part of the vector.

Vectors

A vector is a quantity that has both magnitude and direction.

A vector should tell you how much and which way .

For example, a man is driving his car east along a freeway at $100\phantom{\rule{2pt}{0ex}}\mathrm{km}·\mathrm{h}{}^{-1}$ . What we have given here is a vector – the velocity. The car is moving at $100\phantom{\rule{2pt}{0ex}}\mathrm{km}·\mathrm{h}{}^{-1}$ (this is the magnitude) and we know where it is going – east (this is the direction). Thus, we know the speed and direction of the car. These two quantities, a magnitude and a direction, form a vector we call velocity.

## Notation

Vectors are different to scalars and therefore have their own notation.

## Mathematical representation

There are many ways of writing the symbol for a vector. Vectors are denoted by symbols with an arrow pointing to the right above it. For example, $\stackrel{\to }{a}$ , $\stackrel{\to }{v}$ and $\stackrel{\to }{F}$ represent the vectors acceleration, velocity and force, meaning they have both a magnitude and a direction.

Sometimes just the magnitude of a vector is needed. In this case, the arrow is omitted. In other words, $F$ denotes the magnitude of the vector $\stackrel{\to }{F}$ . $|\stackrel{\to }{F}|$ is another way of representing the magnitude of a vector.

## Graphical representation

Vectors are drawn as arrows. An arrow has both a magnitude (how long it is) and a direction (the direction in which it points). The starting point of a vector is known as the tail and the end point is known as the head .

## Directions

There are many acceptable methods of writing vectors. As long as the vector has a magnitude and a direction, it is most likely acceptable. These different methods come from the different methods of expressing a direction for a vector.

## Relative directions

The simplest method of expressing direction is with relative directions: to the left, to the right, forward, backward, up and down.

## Compass directions

Another common method of expressing directions is to use the points of a compass: North, South, East, and West. If a vector does not point exactly in one of the compass directions, then we use an angle. For example, we can have a vector pointing $40{}^{\circ }$ North of West. Start with the vector pointing along the West direction: Then rotate the vector towards the north until there is a $40{}^{\circ }$ angle between the vector and the West. The direction of this vector can also be described as: W $40{}^{\circ }$ N (West $40{}^{\circ }$ North); or N $50{}^{\circ }$ W (North $50{}^{\circ }$ West)

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
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What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
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anyone know any internet site where one can find nanotechnology papers?
research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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absolutely yes
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how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
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what is fullerene does it is used to make bukky balls
are you nano engineer ?
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fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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