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No requirement for a right triangle

While this example is based on a right triangle, that is not a requirement for adding vectors. I chose a 3-4-5 right triangle for the examplebecause I knew what the answer would be in advance and also because it should have been familiar to you.

Graphic representation of vectors

Many problems in kinematics and kinetics can be solved graphically using a graphical representation of vectors.

Vectors are typically drawn as a heavy line between two points with an arrow head at one end and nothing in particular at the other end. The end with thearrow head is commonly called the head of the vector and the other end is commonly called the tail.

Somewhat inconvenient

Drawing vectors on graph paper can be inconvenient, particularly if you don't have graph paper available. Fortunately, as you willsoon learn, it isn't necessary to use graphics to solve vector problems. You can also solve such problems mathematically, which will often be the better choice.

Addition and subtraction of vectors

There are at least two kinds of quantities in physics:

  • scalars , having magnitude only
  • vectors , having both magnitude and direction

You already know how to do scalar arithmetic. Otherwise, you probably wouldn't be interested in physics.

Another example

Let's go back to our original equation

vecAC = vecAB + vecBC

and assume that the magnitude of vecAB is 30 and the magnitude of vecBC is 40. The sum of those two vectors (vecAC) can have a magnitude ranging anywhere from 10 to 70depending on the relative angles of vecAB and vecBC.

A triangle with sides of 30, 4 0, and ?

Consider the triangle shown in Figure 3 .

Figure 3 - A triangle with sides of 30, 40, and ?.

Missing image

Pretend that instead of walking due north from point B as shown in Figure 2 , you change direction and walk northwest (135 degrees relative to the east-west horizontal line with east being zerodegrees) keeping the length at 40 meters.

What happened to the displacement?

What is the displacement of the point C relative to the point A? I can't do the arithmetic in my head for this problem, but I can measure the length of vecAC to be about 28 meters and the angle of vecAC to be a little less than 90degrees. (I will show you how to write a script to solve this problem mathematically later.)

Any number of displacements can be added

There is no limit to the number of displacement vectors that can be added in this manner. For example, pretend that you walk from point A,

  • 10 meters east to point B,
  • 12 meters southwest to point C,
  • 30 meters north to point D,
  • 15 meters east to point E,
  • 35 meters southwest to point F, and
  • 44 meters north to point G where you stop.

Your displacement vecAG will be

vecAG = vecAB + vecBC + vecCD + vecDE + vecEF + vecFG

even if your zigzag path crosses back over itself one or more times.

Vector diagram for the sum of six vectors

Figure 4 shows the graphical addition of the six vectors described above.

Figure 4 - Vector diagram for the sum of six vectors.

Missing image

The displacement in Figure 4 is the vector with its tail at A and its head at G, which you could measure with a measuring stick anda protractor.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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Source:  OpenStax, Game 2302 - mathematical applications for game development. OpenStax CNX. Jan 09, 2016 Download for free at https://legacy.cnx.org/content/col11450/1.33
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