<< Chapter < Page Chapter >> Page >

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?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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} {} ?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

Questions & Answers

How is the de Broglie wavelength of electrons related to the quantization of their orbits in atoms and molecules?
Larissa Reply
How do you convert 0.0045kgcm³ to the si unit?
EDYKING Reply
how many state of matter do we really have like I mean... is there any newly discovered state of matter?
Falana Reply
I only know 5: •Solids •Liquids •Gases •Plasma •Bose-Einstein condensate
Thapelo
Alright Thank you
Falana
Which one is the Bose-Einstein
James
can you explain what plasma and the I her one you mentioned
Olatunde
u can say sun or stars are just the state of plasma
Mohit
but the are more than seven
Issa
what the meaning of continuum
Akhigbe Reply
What state of matter is fire
Thapelo Reply
fire is not in any state of matter...fire is rather a form of energy produced from an oxidising reaction.
Xenda
Isn`t fire the plasma state of matter?
Walter
all this while I taught it was plasma
Victor
How can you define time?
Thapelo Reply
Time can be defined as a continuous , dynamic , irreversible , unpredictable quantity .
Tanaya
unpredictable? but I can say after one o'clock its going to be two o'clock predictably!
Victor
what is the relativity of physics
Paul Reply
How do you convert 0.0045kgcm³ to the si unit?
flint
What is the formula for motion
Anthony Reply
V=u+at V²=u²-2as
flint
S=ut+½at
flint
they are eqns of linear motion
King
S=Vt
Thapelo
v=u+at s=ut+at^\2 v^=u^+2as where ^=2
King
hi
Mehadi
hello
King
Explain dopplers effect
Jennifer Reply
Not yet learnt
Bob
Explain motion with types
Bob
Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.
Alabi Reply
Scalar quantity Because acceleration has only magnitude
Bob
acleration is vectr quatity it is found in a spefied direction and it is product of displcemnt
bhat
its a scalar quantity
Paul
velocity is speed and direction. since velocity is a part of acceleration that makes acceleration a vector quantity. an example of this is centripetal acceleration. when you're moving in a circular patter at a constant speed, you are still accelerating because your direction is constantly changing.
Josh
acceleration is a vector quantity. As explained by Josh Thompson, even in circular motion, bodies undergoing circular motion only accelerate because on the constantly changing direction of their constant speed. also retardation and acceleration are differentiated by virtue of their direction in
fitzgerald
respect to prevailing force
fitzgerald
What is the difference between impulse and momentum?
Manyo
Momentum is the product of the mass of a body and the change in velocity of its motion. ie P=m(v-u)/t (SI unit is kgm/s). it is literally the impact of collision from a moving body. While Impulse is the product of momentum and time. I = Pt (SI unit is kgm) or it is literally the change in momentum
fitzgerald
Or I = m(v-u)
fitzgerald
Calculation of kinetic and potential energy
dion Reply
K.e=mv² P.e=mgh
Malia
K is actually 1/2 mv^2
Josh
what impulse is given to an a-particle of mass 6.7*10^-27 kg if it is ejected from a stationary nucleus at a speed of 3.2*10^-6ms²? what average force is needed if it is ejected in approximately 10^-8 s?
John
speed=velocity÷time velocity=speed×time=3.2×10^-6×10^-8=32×10^-14m/s impulse [I]=∆momentum[P]=mass×velocity=6.7×10^-27×32×10^-14=214.4×10^-41kg/ms force=impulse÷time=214.4×10^-41÷10^-8=214.4×10^-33N. dats how I solved it.if wrong pls correct me.
Melody
what is sound wave
Nworu Reply
sound wave is a mechanical longitudinal wave that transfers energy from one point to another
Ogor
its a longitudnal wave which is associted wth compresion nad rearfractions
bhat
what is power
PROMISE Reply
it's also a capability to do something or act in a particular way.
Kayode
Newton laws of motion
Mike
power also known as the rate of ability to do work
Slim
power means capabilty to do work p=w/t its unit is watt or j/s it also represents how much work is done fr evry second
bhat
what does fluorine do?
Cheyanne Reply
strengthen and whiten teeth.
Gia
a simple pendulum make 50 oscillation in 1minute, what is it period of oscillation?
Nansing Reply
length of pendulm?
bhat
Practice Key Terms 1

Get the best College physics course in your pocket!





Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask