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Phet explorations: vector addition

Learn how to add vectors. Drag vectors onto a graph, change their length and angle, and sum them together. The magnitude, angle, and components of each vector can be displayed in several formats.

Vector Addition

Summary

  • The analytical method of vector addition and subtraction involves using the Pythagorean theorem and trigonometric identities to determine the magnitude and direction of a resultant vector.
  • The steps to add vectors A size 12{A} {} and B size 12{B} {} using the analytical method are as follows:

    Step 1: Determine the coordinate system for the vectors. Then, determine the horizontal and vertical components of each vector using the equations

    A x = A cos θ B x = B cos θ alignl { stack { size 12{A rSub { size 8{x} } =A"cos"θ} {} #B rSub { size 8{x} } =B"cos"θ {} } } {}

    and

    A y = A sin θ B y = B sin θ . alignl { stack { size 12{A rSub { size 8{y} } =A" sin"θ} {} #B=B suby " sin "θ {} } } {}

    Step 2: Add the horizontal and vertical components of each vector to determine the components R x size 12{R rSub { size 8{x} } } {} and R y size 12{R rSub { size 8{y} } } {} of the resultant vector, R size 12{R} {} :

    R x = A x + B x size 12{R rSub { size 8{x} } =A rSub { size 8{x} } +B rSub { size 8{x} } } {}

    and

    R y = A y + B y . size 12{R rSub { size 8{y} } =A rSub { size 8{y} } +B rSub { size 8{y} } } {}

    Step 3: Use the Pythagorean theorem to determine the magnitude, R size 12{R} {} , of the resultant vector R size 12{R} {} :

    R = R x 2 + R y 2 . size 12{R= sqrt {R rSub { size 8{x} } rSup { size 8{2} } +R rSub { size 8{y} } rSup { size 8{2} } } } {}

    Step 4: Use a trigonometric identity to determine the direction, θ size 12{θ} {} , of R size 12{R} {} :

    θ = tan 1 ( R y / R x ) . size 12{θ="tan" rSup { size 8{ - 1} } \( R rSub { size 8{y} } /R rSub { size 8{x} } \) } {}

Conceptual questions

Suppose you add two vectors A size 12{A} {} and B size 12{B} {} . What relative direction between them produces the resultant with the greatest magnitude? What is the maximum magnitude? What relative direction between them produces the resultant with the smallest magnitude? What is the minimum magnitude?

Give an example of a nonzero vector that has a component of zero.

Explain why a vector cannot have a component greater than its own magnitude.

If the vectors A size 12{A} {} and B size 12{B} {} are perpendicular, what is the component of A size 12{A} {} along the direction of B size 12{B} {} ? What is the component of B size 12{B} {} along the direction of A size 12{A} {} ?

Problems&Exercises

Find the following for path C in [link] : (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.

A map of city is shown. The houses are in form of square blocks of side one hundred and twenty meter each. Four paths A B C and D are shown in different colors. The path c shown as blue extends to one block towards north, then five blocks towards east and then two blocks towards south then one block towards west and one block towards north and finally three blocks towards west. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish for path C.
The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side.

(a) 1.56 km

(b) 120 m east

Find the following for path D in [link] : (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.

Find the north and east components of the displacement from San Francisco to Sacramento shown in [link] .

A map of northern California with a circle with a radius of one hundred twenty three kilometers centered on San Francisco. Sacramento lies on the circumference of this circle in a direction forty-five degrees north of east from San Francisco.

North-component 87.0 km, east-component 87.0 km

Solve the following problem using analytical techniques: 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 the given figure displacement of a person is shown. First movement of the person is shown as vector A from origin along negative x axis. He then turns to his right. His movement is now shown as a vertical vector in north direction. The displacement vector R is also shown. In the question you are asked to find the displacement of the person from the start to finish.
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} {} .

Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique.

Repeat [link] using analytical techniques, but reverse the order of the two legs of the walk and show that you get the same final result. (This problem shows that adding them in reverse order gives the same result—that is, B + A = A + B .) Discuss how taking another path to reach the same point might help to overcome an obstacle blocking you other path.

30.8 m, 35.8 west of north

You drive 7 . 50 km size 12{7 "." "50 km"} {} in a straight line in a direction 15º size 12{"15º"} {} east of north. (a) Find the distances you would have to drive straight east and then straight north to arrive at the same point. (This determination is equivalent to find the components of the displacement along the east and north directions.) (b) Show that you still arrive at the same point if the east and north legs are reversed in order.

Do [link] again using analytical techniques and change the second leg of the walk to 25.0 m straight south. (This is equivalent to subtracting B size 12{B} {} from A size 12{A} {} —that is, finding R = A – B ) (b) Repeat again, but now you first walk 25 . 0 m size 12{"25" "." "0 m"} {} north and then 18 . 0 m size 12{"18" "." "0 m"} {} east. (This is equivalent to subtract A size 12{A} {} from B size 12{B} {} —that is, to find A = B + C size 12{A=B+C} {} . Is that consistent with your result?)

(a) 30 . 8 m size 12{"30" "." "8 m"} {} , 54 . size 12{"54" "." 2°} {} south of west

(b) 30 . 8 m size 12{"30" "." "8 m"} {} , 54 . size 12{"54" "." 2°} {} north of east

A new landowner has a triangular piece of flat land she wishes to fence. Starting at the west corner, she measures the first side to be 80.0 m long and the next to be 105 m. These sides are represented as displacement vectors A size 12{A} {} from B size 12{B} {} in [link] . She then correctly calculates the length and orientation of the third side C size 12{C} {} . What is her result?

In the given figure the sides of a triangular piece of land are shown in vector form. West corner is at origin. A vector starts from the origin towards south east direction and makes an angle twenty-one degrees with the horizontal. Then from the head of this vector another vector B making an angle eleven degrees with the vertical is drawn upwards. Then another vector C from the head of the vector B to the tail of the initial vector is drawn. The length and orientation of side C is indicated as unknown, represented by a question mark.

You fly 32 . 0 km size 12{"32" "." "0 km"} {} in a straight line in still air in the direction 35.0º size 12{"35"°} {} south of west. (a) Find the distances you would have to fly straight south and then straight west to arrive at the same point. (This determination is equivalent to finding the components of the displacement along the south and west directions.) (b) Find the distances you would have to fly first in a direction 45.0º size 12{"45.0º"} {} south of west and then in a direction 45.0º size 12{"45.0º"} {} west of north. These are the components of the displacement along a different set of axes—one rotated 45º size 12{"45"°} {} .

18.4 km south, then 26.2 km west(b) 31.5 km at 45.0º size 12{"45.0º"} {} south of west, then 5.56 km at 45.0º size 12{"45.0º"} {} west of north

A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A , size 12{A,} {} B , size 12{B,} {} and C size 12{C} {} in [link] , and then correctly calculates the length and orientation of the fourth side D size 12{D} {} . What is his result?

A quadrilateral with sides A, B, C, and D. A begins at the end of D and is 4 point seven zero kilometers  at an angle of 7 point 5 degrees south of west. B begins at the end of A and is 2 point four eight kilometers in a direction sixteen degrees west of north. C begins at the end of B and is 3 point zero 2 kilometers in a direction nineteen degrees north of west. D begins at the end of C and runs distance and direction that must be calculated

In an attempt to escape his island, Gilligan builds a raft and sets to sea. The wind shifts a great deal during the day, and he is blown along the following straight lines: 2 . 50 km size 12{2 "." "50 km"} {} 45.0º size 12{"45.0º"} {} north of west; then 4 . 70 km size 12{4 "." "70 km"} {} 60.0º size 12{"60"°} {} south of east; then 1.30 km size 12{"1.30" " km"} {} 25.0º size 12{"25"°} {} south of west; then 5 . 10 km size 12{5 "." "10 km"} {} straight east; then 1.70 km size 12{"1.70" " km"} {} 5.00º size 12{5 rSup { size 8{ circ } } } {} east of north; then 7 . 20 km size 12{7 "." "20 km"} {} 55.0º size 12{"55"°} {} south of west; and finally 2 . 80 km size 12{2 "." "80 km"} {} 10.0º size 12{"10"°} {} north of east. What is his final position relative to the island?

7 . 34 km size 12{2 "." "97 km"} {} , 63 . size 12{"22" "." 2°} {} south of east

Suppose a pilot flies 40 . 0 km size 12{"40" "." "0 km"} {} in a direction 60º size 12{"60"°} {} north of east and then flies 30 . 0 km size 12{"30" "." "0 km"} {} in a direction 15º size 12{"15"°} {} north of east as shown in [link] . Find her total distance R size 12{R} {} from the starting point and the direction θ size 12{θ} {} of the straight-line path to the final position. Discuss qualitatively how this flight would be altered by a wind from the north and how the effect of the wind would depend on both wind speed and the speed of the plane relative to the air mass.

A triangle  defined by vectors A, B, and R. A begins at the origin and run forty kilometers in a direction sixty degrees north of east. B begins at the end of A and runs thirty kilometers in a direction fifteen degrees north of east. R is the resultant vector and runs from the origin (the beginning of A) to the end of B for a distance and in a direction theta that need to be calculated.

Questions & Answers

what is Nano technology ?
Bob Reply
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?
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?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
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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
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Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Cc test coll. OpenStax CNX. Dec 15, 2015 Download for free at http://legacy.cnx.org/content/col11717/1.4
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