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OB = ( P 2 + Q 2 ) tan α = Q P

These results for vectors at right angle are exactly same as determined, using Pythagoras theorem.

Problem : Three radial vectors OA, OB and OC act at the center of a circle of radius “r” as shown in the figure. Find the magnitude of resultant vector.

Sum of three vectors

Three radial vectors OA, OB and OC act at the center of a circle of radius “r”.

Solution : It is evident that vectors are equal in magnitude and is equal to the radius of the circle. The magnitude of the resultant of horizontal and vertical vectors is :

R’ = ( r 2 + r 2 ) = 2 r

The resultant of horizontal and vertical vectors is along the bisector of angle i.e. along the remaining third vector OB. Hence, magnitude of resultant of all three vectors is :

R’ = OB + R’ = r + 2 r = ( 1 + 2 ) r

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Problem : At what angle does two vectors a + b and a - b act so that the resultant is ( 3 a 2 + b 2 ) .

Solution : The magnitude of resultant of two vectors is given by :

Angle

The angle between the sum and difference of vectors.

R = { ( a + b ) 2 + ( a - b ) 2 + 2 ( a + b ) ( a - b ) cos θ }

Substituting the expression for magnitude of resultant as given,

( 3 a 2 + b 2 ) = { ( a + b ) 2 + ( a - b ) 2 + 2 ( a + b ) ( a - b ) cos θ }

Squaring on both sides, we have :

( 3 a 2 + b 2 ) = { ( a + b ) 2 + ( a - b ) 2 + 2 ( a + b ) ( a - b ) cos θ }

cos θ = ( a 2 - b 2 ) 2 ( a 2 - b 2 ) = 1 2 = cos 60 °

θ = 60 °

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Nature of vector addition

Vector sum and difference

The magnitude of sum of two vectors is either less than or equal to sum of the magnitudes of individual vectors. Symbolically, if a and b be two vectors, then

| a + b | | a | + | b |

We know that vectors a , b and their sum a + b is represented by three side of a triangle OAC. Further we know that a side of triangle is always less than the sum of remaining two sides. It means that :

Two vectors

Sum of two vectors

OC < OA + AC OC < OA + OB | a + b | < | a | + | b |

There is one possibility, however, that two vectors a and b are collinear and act in the same direction. In that case, magnitude of their resultant will be "equal to" the sum of the magnitudes of individual vector. This magnitude represents the maximum or greatest magnitude of two vectors being combined.

OC = OA + OB | a + b | = | a | + | b |

Combining two results, we have :

| a + b | | a | + | b |

On the other hand, the magnitude of difference of two vectors is either greater than or equal to difference of the magnitudes of individual vectors. Symbolically, if a and b be two vectors, then

| a - b | | a | - | b |

We know that vectors a , b and their difference a - b are represented by three side of a triangle OAE. Further we know that a side of triangle is always less than the sum of remaining two sides. It means that sum of two sides is greater than the third side :

Two vectors

Difference of two vectors

OE + AE > OA OE > OA - AE | a - b | > | a | - | b |

There is one possibility, however, that two vectors a and b are collinear and act in the opposite directions. In that case, magnitude of their difference will be equal to the difference of the magnitudes of individual vector. This magnitude represents the minimum or least magnitude of two vectors being combined.

OE = OA - AE | a - b | = | a | - | b |

Combining two results, we have :

| a - b | | a | - | b |

Lami's theorem

Lami's theorem relates magnitude of three non-collinear vectors with the angles enclosed between pair of two vectors, provided resultant of three vectors is zero (null vector). This theorem is a manifestation of triangle law of addition. According to this theorem, if resultant of three vectors a , b and c is zero (null vector), then

a sin α = b sin β = c sin γ

Three vectors

Three non-collinear vectors.

where α, β and γ be the angle between the remaining pairs of vectors.

We know that if the resultant of three vectors is zero, then they are represented by three sides of a triangle in magnitude and direction.

Three vectors

Three vectors are represented by three sides of a triangle.

Considering the magnitude of vectors and applying sine law of triangle, we have :

AB sin BCA = BC sin CAB = CA sin ABC

AB sin ( π - α ) = BC sin ( π - β ) = CA sin ( π - γ )

AB sin α = BC sin β = CA sin γ

It is important to note that the ratio involves exterior (outside) angles – not the interior angles of the triangle. Also, the angle associated with the magnitude of a vector in the individual ratio is the included angle between the remaining vectors.

Exercises

Two forces of 10 N and 25 N are applied on a body. Find the magnitude of maximum and minimum resultant force.

Resultant force is maximum when force vectors act along the same direction. The magnitude of resultant force under this condition is :

R max = 10 + 25 = 35 N R min = 25 - 10 = 15 N

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Can a body subjected to three coplanar forces 5 N, 17 N and 9 N be in equilibrium?

The resultant force, on a body in equilibrium, is zero. It means that three forces can be represented along three sides of a triangle. However, we know that sum of any two sides is greater than third side. In this case, we see that :

5 + 9 < 17

Clearly, three given forces can not be represented by three sides of a triangle. Thus, we conclude that the body is not in equilibrium.

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Under what condition does the magnitude of the resultant of two vectors of equal magnitude, is equal in magnitude to either of two equal vectors?

We know that resultant of two vectors is represented by the closing side of a triangle. If the triangle is equilateral then all three sides are equal. As such magnitude of the resultant of two vectors is equal to the magnitude of either of the two vectors.

Two vectors

Resultant of two vectors

Under this condition, vectors of equal magnitude make an angle of 120° between them.

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Questions & Answers

what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
n=a+b/T² find the linear express
Donsmart Reply
Quiklyyy
Sultan Reply
Moment of inertia of a bar in terms of perpendicular axis theorem
Sultan Reply
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Yara Reply
Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
Rama Reply
A balloon is released from the ground which rises vertically up with acceleration 1.4m/sec^2.a ball is released from the balloon 20 second after the balloon has left the ground. The maximum height reached by the ball from the ground is
Lucky Reply
work done by frictional force formula
Sudeer Reply
Torque
Misthu Reply
Why are we takingspherical surface area in case of solid sphere
Saswat Reply
In all situatuons, what can I generalize?
Cart Reply
the body travels the distance of d=( 14+- 0.2)m in t=( 4.0 +- 0.3) s calculate it's velocity with error limit find Percentage error
Clinton Reply
Explain it ?Fy=?sN?mg=0?N=mg?s
Admire Reply

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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