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Learning objectives

By the end of this section, you will be able to:

  • Understand the rules of vector addition and subtraction using analytical methods.
  • Apply analytical methods to determine vertical and horizontal component vectors.
  • Apply analytical methods to determine the magnitude and direction of a resultant vector.

The information presented in this section supports the following AP® learning objectives and science practices:

  • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2)

Analytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made. Analytical methods are limited only by the accuracy and precision with which physical quantities are known.

Resolving a vector into perpendicular components

Analytical techniques and right triangles go hand-in-hand in physics because (among other things) motions along perpendicular directions are independent. We very often need to separate a vector into perpendicular components. For example, given a vector like A size 12{A} {} in [link] , we may wish to find which two perpendicular vectors, A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} , add to produce it.

In the given figure a dotted vector A sub x is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A sub y at an angle theta from the x axis. On the graph a vector A, inclined at an angle theta with x axis is shown. Therefore vector A is the sum of the vectors A sub x and A sub y.
The vector A size 12{A} {} , with its tail at the origin of an x , y -coordinate system, is shown together with its x - and y -components, A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} . These vectors form a right triangle. The analytical relationships among these vectors are summarized below.

A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} are defined to be the components of A size 12{A} {} along the x - and y -axes. The three vectors A size 12{A} {} , A x size 12{A rSub { size 8{x} } } {} , and A y size 12{A rSub { size 8{y} } } {} form a right triangle:

A x  + A y  = A . size 12{A rSub { size 8{x} } bold " + A" rSub { size 8{y} } bold " = A."} {}

Note that this relationship between vector components and the resultant vector holds only for vector quantities (which include both magnitude and direction). The relationship does not apply for the magnitudes alone. For example, if A x = 3 m size 12{A rSub { size 8{x} } } {} east, A y = 4 m size 12{A rSub { size 8{y} } } {} north, and A = 5 m size 12{A} {} north-east, then it is true that the vectors A x  + A y  = A size 12{A rSub { size 8{x} } bold " + A" rSub { size 8{y} } bold " = A"} {} . However, it is not true that the sum of the magnitudes of the vectors is also equal. That is,

3 m + 4 m   5 m alignl { stack { size 12{"3 M + 4 M "<>" 5 M"} {} # {}} } {}

Thus,

A x + A y A size 12{A rSub { size 8{x} } +A rSub { size 8{y} }<>A} {}

If the vector A size 12{A} {} is known, then its magnitude A size 12{A} {} (its length) and its angle θ size 12{θ} {} (its direction) are known. To find A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} , its x - and y -components, we use the following relationships for a right triangle.

A x = A cos θ size 12{A rSub { size 8{x} } =A"cos"θ} {}

and

A y = A sin θ . size 12{A rSub { size 8{y} } =A"sin"θ"."} {}
]A dotted vector A sub x whose magnitude is equal to A cosine theta is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y whose magnitude is equal to A sine theta is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A-y at an angle theta from the x axis. Therefore vector A is the sum of the vectors A sub x and A sub y.
The magnitudes of the vector components A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} can be related to the resultant vector A size 12{A} {} and the angle θ size 12{θ} {} with trigonometric identities. Here we see that A x = A cos θ size 12{A rSub { size 8{x} } =A"cos"θ} {} and A y = A sin θ size 12{A rSub { size 8{y} } =A"sin"θ} {} .

Suppose, for example, that A size 12{A} {} is the vector representing the total displacement of the person walking in a city considered in Kinematics in Two Dimensions: An Introduction and Vector Addition and Subtraction: Graphical Methods .

In the given figure a vector A of magnitude ten point three blocks is inclined at an angle twenty nine point one degrees to the positive x axis. The horizontal component A sub x of vector A is equal to A cosine theta which is equal to ten point three blocks multiplied to cosine twenty nine point one degrees which is equal to nine blocks east. Also the vertical component A sub y of vector A is equal to A sin theta is equal to ten point three blocks multiplied to sine twenty nine point one degrees,  which is equal to five point zero blocks north.
We can use the relationships A x = A cos θ size 12{A rSub { size 8{x} } =A"cos"θ} {} and A y = A sin θ size 12{A rSub { size 8{y} } =A"sin"θ} {} to determine the magnitude of the horizontal and vertical component vectors in this example.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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
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Source:  OpenStax, Sample chapters: openstax college physics for ap® courses. OpenStax CNX. Oct 23, 2015 Download for free at http://legacy.cnx.org/content/col11896/1.9
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