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A = A x 2 + A y 2 size 12{A= sqrt {A rSub { size 8{x} rSup { size 8{2} } } +A rSub { size 8{y} rSup { size 8{2} } } } } {}
θ = tan 1 ( A y / A x ) . size 12{θ="tan" rSup { size 8{ - 1} } \( A rSub { size 8{y} } /A rSub { size 8{x} } \) } {}
Vector A is shown with its horizontal and vertical components A sub x and A sub y respectively. The magnitude of vector A is equal to the square root of A sub x squared plus A sub y squared. The angle theta of the vector A with the x axis is equal to inverse tangent of A sub y over A sub x
The magnitude and direction of the resultant vector can be determined once the horizontal and vertical components A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} have been determined.

Note that the equation A = A x 2 + A y 2 size 12{A= sqrt {A rSub { size 8{x} rSup { size 8{2} } } +A rSub { size 8{y} rSup { size 8{2} } } } } {} is just the Pythagorean theorem relating the legs of a right triangle to the length of the hypotenuse. For example, if A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} are 9 and 5 blocks, respectively, then A = 9 2 +5 2 =10 . 3 size 12{A= sqrt {9 rSup { size 8{2} } "+5" rSup { size 8{2} } } "=10" "." 3} {} blocks, again consistent with the example of the person walking in a city. Finally, the direction is θ = tan –1 ( 5/9 ) =29.1º size 12{θ="tan" rSup { size 8{–1} } \( "5/9" \) "=29" "." 1 rSup { size 8{o} } } {} , as before.

Determining vectors and vector components with analytical methods

Equations 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} {} are used to find the perpendicular components of a vector—that is, to go from A size 12{A} {} and θ size 12{θ} {} to A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} . Equations A = A x 2 + A y 2 size 12{A= sqrt {A rSub { size 8{x} rSup { size 8{2} } } +A rSub { size 8{y} rSup { size 8{2} } } } } {} and θ = tan –1 ( A y / A x ) are used to find a vector from its perpendicular components—that is, to go from A x and A y to A and θ . Both processes are crucial to analytical methods of vector addition and subtraction.

Adding vectors using analytical methods

To see how to add vectors using perpendicular components, consider [link] , in which the vectors A size 12{A} {} and B size 12{B} {} are added to produce the resultant R size 12{R} {} .

Two vectors A and B are shown. The tail of vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown.
Vectors A size 12{A} {} and B size 12{B} {} are two legs of a walk, and R size 12{R} {} is the resultant or total displacement. You can use analytical methods to determine the magnitude and direction of R size 12{R} {} .

If A and B represent two legs of a walk (two displacements), then R is the total displacement. The person taking the walk ends up at the tip of R . There are many ways to arrive at the same point. In particular, the person could have walked first in the x -direction and then in the y -direction. Those paths are the x - and y -components of the resultant, R x and R y size 12{R rSub { size 8{y} } } {} . If we know R x and R y size 12{R rSub { size 8{y} } } {} , we can find R and θ using the equations A = A x 2 + A y 2 and θ = tan –1 ( A y / A x ) size 12{θ="tan" rSup { size 8{–1} } \( A rSub { size 8{y} } /A rSub { size 8{x} } \) } {} . When you use the analytical method of vector addition, you can determine the components or the magnitude and direction of a vector.

Step 1. Identify the x- and y-axes that will be used in the problem. Then, find the components of each vector to be added along the chosen perpendicular axes . Use the equations 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 find the components. In [link] , these components are A x size 12{A rSub { size 8{x} } } {} , A y size 12{A rSub { size 8{y} } } {} , B x size 12{B rSub { size 8{x} } } {} , and B y size 12{B rSub { size 8{y} } } {} . The angles that vectors A size 12{A} {} and B size 12{B} {} make with the x -axis are θ A size 12{θ rSub { size 8{A} } } {} and θ B size 12{θ rSub { size 8{B} } } {} , respectively.

Two vectors A and B are shown. The tail of the vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. The horizontal and vertical components of the vectors A and B are shown with the help of dotted lines. The vectors labeled as A sub x and A sub y are the components of vector A, and B sub x and B sub y as the components of vector B..
To add vectors A size 12{A} {} and B size 12{B} {} , first determine the horizontal and vertical components of each vector. These are the dotted vectors A x size 12{A rSub { size 8{x} } } {} , A y size 12{A rSub { size 8{y} } } {} , B x size 12{B rSub { size 8{x} } } {} and B y size 12{B rSub { size 8{y} } } {} shown in the image.

Step 2. Find the components of the resultant along each axis by adding the components of the individual vectors along that axis . That is, as shown in [link] ,

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} } } {}
Two vectors A and B are shown. The tail of vector B is at the head of vector A and the tail of the vector A is at origin. Both the vectors are in the first quadrant. The resultant R of these two vectors extending from the tail of vector A to the head of vector B is also shown. The vectors A and B are resolved into the horizontal and vertical components shown as dotted lines parallel to x axis and y axis respectively. The horizontal components of vector A and vector B are labeled as A sub x and B sub x and the horizontal component of the resultant R is labeled at R sub x and is equal to A sub x plus B sub x. The vertical components of vector A and vector B are labeled as A sub y and B sub y and the vertical components of the resultant R is labeled as R sub y is equal to A sub y plus B sub y.
The magnitude of the vectors A x size 12{A rSub { size 8{x} } } {} and B x size 12{B rSub { size 8{x} } } {} add to give the magnitude R x size 12{R rSub { size 8{x} } } {} of the resultant vector in the horizontal direction. Similarly, the magnitudes of the vectors A y size 12{A rSub { size 8{y} } } {} and B y size 12{B rSub { size 8{y} } } {} add to give the magnitude R y size 12{R rSub { size 8{y} } } {} of the resultant vector in the vertical direction.

Components along the same axis, say the x -axis, are vectors along the same line and, thus, can be added to one another like ordinary numbers. The same is true for components along the y -axis. (For example, a 9-block eastward walk could be taken in two legs, the first 3 blocks east and the second 6 blocks east, for a total of 9, because they are along the same direction.) So resolving vectors into components along common axes makes it easier to add them. Now that the components of R size 12{R} {} are known, its magnitude and direction can be found.

Questions & Answers

tree physical properties of heat
Bello Reply
tree is a type of organism that grows very tall and have a wood trunk and branches with leaves... how is that related to heat? what did you smoke man?
what are the uses of dimensional analysis
Racheal Reply
Dimensional Analysis. The study of relationships between physical quantities with the help of their dimensions and units of measurements is called dimensional analysis. We use dimensional analysis in order to convert a unit from one form to another.
Emmanuel
meaning of OE and making of the subscript nc
ferunmi Reply
can I ask a question
Negash
kinetic functional force
Moyagabo Reply
what is a principal wave?
Haider Reply
A wave the movement of particles on rest position transferring energy from one place to another
Gabche
not wave. i need to know principal wave or waves.
Haider
principle wave is a superposition of wave when two or more waves meet at a point , whose amplitude is the algebraic sum of the amplitude of the waves
arshad
kindly define principal wave not principle wave (principle of super position) if u can understand my question
Haider
what is a model?
Ella Reply
hi
Muhanned
why are electros emitted only when the frequency of the incident radiation is greater than a certain value
ANSELEM Reply
b/c u have to know that for emission of electron need specific amount of energy which are gain by electron for emission . if incident rays have that amount of energy electron can be emitted, otherwise no way.
Nazir
search photoelectric effect on Google
Nazir
what is ohm's law
Pamilerin Reply
states that electric current in a given metallic conductor is directly proportional to the potential difference applied between its end, provided that the temperature of the conductor and other physical factors such as length and cross-sectional area remains constant. mathematically V=IR
ANIEFIOK
hi
Gundala
A body travelling at a velocity of 30ms^-1 in a straight line is brought to rest by application of brakes. if it covers a distance of 100m during this period, find the retardation.
Pamilerin Reply
just use v^2-u^2=2as
Gundala
how often does electrolyte emits?
alhassan
just use +€^3.7°√π%-4¢•∆¥%
v^2-u^2=2as v=0,u=30,s=100 -30^2=2a*100 -900=200a a=-900/200 a=-4.5m/s^2
akinyemi
what is distribution of trade
Grace Reply
what's acceleration
Joshua Reply
The change in position of an object with respect to time
Mfizi
Acceleration is velocity all over time
Pamilerin
hi
Stephen
It's not It's the change of velocity relative to time
Laura
Velocity is the change of position relative to time
Laura
acceleration it is the rate of change in velocity with time
Stephen
acceleration is change in velocity per rate of time
Noara
what is ohm's law
Stephen
Ohm's law is related to resistance by which volatge is the multiplication of current and resistance ( U=RI)
Laura
acceleration is the rate of change. of displacement with time.
Radical
the rate of change of velocity is called acceleration
Asma
how i don understand
Willam Reply
how do I access the Multiple Choice Questions? the button never works and the essay one doesn't either
Savannah Reply
How do you determine the magnitude of force
Peace Reply
mass × acceleration OR Work done ÷ distance
Seema
Which eye defect is corrected by a lens having different curvatures in two perpendicular directions?
Valentina Reply
acute astigmatism?
Practice Key Terms 1

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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