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Explain why it is not possible to add a scalar to a vector.

If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more?

Problems&Exercises

Use graphical methods to solve these problems. You may assume data taken from graphs is accurate to three digits.

Find the following for path A in [link] : (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

A map of city is shown. The houses are in form of square blocks of side one hundred and twenty meters each. The path of A extends to three blocks towards north and then one block towards east. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish.
The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side.

(a) 480 m size 12{"480 m"} {}

(b) 379 m size 12{"379 m"} {} , 18.4° size 12{"18" "." "4° east of north"} {} east of north

Find the following for path B in [link] : (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

Find the north and east components of the displacement for the hikers shown in [link] .

north component 3.21 km, east component 3.83 km

Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A size 12{A} {} and B size 12{B} {} , as in [link] , then this problem asks you to find their sum R = A + B size 12{"R = A + B"} {} .)

In this figure coordinate axes are shown. Vector A from the origin towards the negative of x axis is shown. From the head of the vector A another vector B is drawn towards the positive direction of y axis. The resultant R of these two vectors is shown as a vector from the tail of vector A to the head of vector B. This vector R is inclined at an angle theta with the negative x axis.
The two displacements A size 12{A} {} and B size 12{B} {} add to give a total displacement R size 12{R} {} having magnitude R size 12{R} {} and direction θ size 12{θ} {} .

Suppose you first walk 12.0 m in a direction 20° size 12{"20" { size 12{°} } } {} west of north and then 20.0 m in a direction 40.0° size 12{"40" { size 12{°} } } {} south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A size 12{A} {} and B size 12{B} {} , as in [link] , then this problem finds their sum R = A + B size 12{ bold "R = A + B"} {} .)

In the given figure coordinates axes are shown. Vector A with tail at origin is inclined at an angle of twenty degrees with the positive direction of x axis. The magnitude of vector A is twelve meters. Another vector B is starts from the head of vector A and inclined at an angle of forty degrees with the horizontal. The resultant R of the vectors A and B is also drawn from the tail of vector A to the head of vector B. The inclination of vector R is theta with the horizontal.

19 . 5 m size 12{"19" "." "5 m"} {} , 4 . 65° size 12{4 "." "65°"} {} south of west

Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg B size 12{B} {} , which is 20.0 m in a direction exactly 40° size 12{"20" { size 12{°} } } {} south of west, and then leg A size 12{A} {} , which is 12.0 m in a direction exactly 20° size 12{"20" { size 12{°} } } {} west of north. (This problem shows that A + B = B + A size 12{A+B=B+A} {} .)

(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction 40.0° size 12{"40.0" { size 12{°} } } north of east (which is equivalent to subtracting B size 12{B} {} from A size 12{A} {} —that is, to finding R = A B size 12{ bold "R'"=A - B} {} ). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction 40.0° size 12{"40.0" { size 12{°} } } south of west and then 12.0 m in a direction 20.0° size 12{"20.0" { size 12{ ° } } } {} east of south (which is equivalent to subtracting A size 12{A} {} from B size 12{B} {} —that is, to finding R ′′ = B - A = - R size 12{R''= B – A = -R' } {} ). Show that this is the case.

(a) 26 . 6 m size 12{"26" "." "6 m"} {} , 65 . size 12{"65" "." "1°"} {} north of east

(b) 26 . 6 m size 12{"26" "." "6 m"} {} , 65 . size 12{"65" "." "1°"} {} south of west

Show that the order of addition of three vectors does not affect their sum. Show this property by choosing any three vectors A size 12{A} {} , B size 12{B} {} , and C size 12{C} {} , all having different lengths and directions. Find the sum A + B + C size 12{ bold "A + B + C"} {} then find their sum when added in a different order and show the result is the same. (There are five other orders in which A size 12{A} {} , B size 12{B} {} , and C size 12{C} {} can be added; choose only one.)

Show that the sum of the vectors discussed in [link] gives the result shown in [link] .

52 . 9 m size 12{"52" "." "9 m"} {} , 90 . size 12{"90" "." "1°"} {} with respect to the x -axis.

Find the magnitudes of velocities v A size 12{v rSub { size 8{A} } } {} and v B size 12{v rSub { size 8{B} } } {} in [link]

On the graph velocity vector V sub A begins at the origin and is inclined to x axis at an angle of twenty two point five degrees. From the head of vector V sub A another vector V sub B begins. The resultant of the two vectors, labeled V sub tot, is inclined to vector V sub A at twenty six point five degrees and to the vector V sub B at twenty three point zero degrees. V sub tot has a magnitude of 6.72 meters per second.
The two velocities v A size 12{v rSub { size 8{A} } } {} and v B size 12{v rSub { size 8{B} } } {} add to give a total v tot size 12{v rSub { size 8{"tot"} } } {} .

Find the components of v tot size 12{v rSub { size 8{"tot"} } } {} along the x - and y -axes in [link] .

x -component 4.41 m/s

y -component 5.07 m/s

Find the components of v tot size 12{v rSub { size 8{"tot"} } } {} along a set of perpendicular axes rotated 30° size 12{"30°"} {} counterclockwise relative to those in [link] .

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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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