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VECTORS

1. Scalar : A physical quantity having only magnitude but not associated with any direction is called a scalar.
eg: time, mass, distance, speed, work, energy, power, pressure, temperature, electric current, gravitational potential, pole strength, magnetic flux, entropy, electric capacity, velocity of light, large angular displacement, electric charge, etc.
2. Scalars are added and subtracted by algebraic method.
3. Vector : A physical quantity having magnitude as well as associated direction and which obeys vector laws is called a vector.
eg: displacement, velocity, acceleration, force, momentum, impulse, moment of force, small angular displacement, angular velocity, angular acceleration, magnetic moment, dipole moment, current density, intensity of electric field or magnetic field, shearing stress, weight, centrifugal force, infinitesimally small area, etc.
4. Vectors are completely described by a number with a unit followed by a statement of direction.
5. Angle can be considered as vector if it is small. Large angles can not be treated as vectors as they do not obey laws of vector addition.

6. Surface area can be treated both as a scalar and a vector. A is magnitude of surface area which is a scalar. This area is enclosed by a closed curve as shown if  is a unit vector normal to the surface, we can write  as a vector.
 Surface area is a vector.
[If the four fingers of right hand curl along the direction of arrow of enclosing curve, thumb indicates direction of area vector]

7. Tensor is a physical quantity which will have different values along different directions.
 e.g. Moment of inertia, stress.
8. A vector is represented by a directed line segment. The length of the line segment is proportional to the magnitude of the vector.
9. The magnitude or modulus of a vector (|| or r) is a scalar.
10. Electric current, velocity of light has both magnitude and direction but they do not obey the laws of vector addition. Hence they are scalars.
11. Equal vectors : Two vectors are said to be equal if they have the same magnitude and direction irrespective of their initial points.
12. Negative vectors :  and   are vectors having the same magnitude and opposite direction. is called the negative of .
13. Proper vector : A vector whose magnitude is not zero is known as proper vector.
14. Null Vector (Zero Vector): It is a vector whose magnitude is zero and direction is unspecified.
Examples:
a) Displacement after one complete revolution.
b) Velocity of vertically projected body at the highest point
15. Parallel vectors : Vectors in the same direction are called parallel vectors.
16. Antiparallel vectors : Vectors in opposite direction are called antiparallel vectors.
17. Like vectors or co-directional vectors : The vectors directed in the same direction, irrespective of their magnitudes are called co-directional vectors or like vectors.
18. Collinear vectors : Two or more vectors parallel or antiparallel to each other are called collinear vectors.
19. Coplanar vectors : Vectors lying on the same plane are called coplanar vectors and the plane in which they lie is called the plane of the vectors. 
20. Unit vector : It is a vector whose magnitude is unity. A unit vector parallel to a given vector.
21. If is a vector, the unit vector in the direction of is written as.
are units vectors along x, y and z axis.
22. Position vector : The vector which is used to specify the position of a point ‘P’ with respect to some fixed point ‘O’ is represented by  and is known as the position vector of ‘P’ with respect to ‘O’. 
23. Real Vector or Polar Vector: If the direction of a vector is independent of the coordinate system., then it is called a polar vector. Example: linear velocity, linear momentum, force, etc.,
24. Pseudo or axial vectors : Axial vectors or pseudo vectors are those whose direction is fixed by convention and reverses in a mirror reflection. Cross product of two vectors gives an axial vector. eg : Torque, angular velocity, etc.
25. A vector remains unchanged when it is moved parallel to itself.
26. If m is a scalar and  a vector, then m  is a vector. Its magnitude is m times that of magnitude of . Its direction is the same as that of  , if m is positive and opposite if m is negative,

. If m is zero, ma null vector.
27. Vector multiplication obeys commutative law when multiplied by a scalar.
s = s where s is scalar.


28. Vector multiplication obeys associative law when multiplier by a scalar

29. Vector multiplication obeys distributive law when multiplied by a scalar. s(+)
=s+s.


ADDITION OF VECTORS:
30. Addition of vectors is also called resultant of vectors.
31. Resultant is a single vector that gives the total effect of number of vectors.
Resultant can be found by using
a) Triangle law of vectors
b) Parallelogram law of vectors
c) Polygon law of vectors
32. Two vectors can be added either by triangle law or parallelogram law of vectors.
33. Triangle law : If two vectors are represented in magnitude and direction by the two sides of a triangle taken in order, then the third side taken in the reverse order represents their sum or resultant in magnitude as well as in direction.

34. If two vectors  are represented by the two sides of a parallelogram drawn from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.
R =

35. The resultant of two vectors is the vectorial addition of two vectors.
36. The resultant of any two vectors makes lesser angle with the greater vector.

 

37. If   

38. The magnitude of the resultant of two vectors of magnitudes a and b with arbitrary directions must be in the range (a – b) to (a + b).
39.    are two vectors which when added give a vector and if
     i)  are parallel vectors (  =  0o)
     ii) are
          perpendicular vectors (= 90o)
     iii)  are antiparallel
            vectors (= 180o)
     iv)  are inclined to 
           each other at 120o

v) If  .
vi) If  .
vii) If  
40. If two vectors each of magnitude F act at a point, the magnitude of their resultant (R) depends on the anglebetween them.
R = 2Fcos(θ/2).

Angle between forces () Magnitude of resultant
0° 2F
60° F
90° F
120° F°
180° 0

 

41. Minimum number of equal vectors to give a zero resultant is 2.
42. The minimum number of unequal vectors to give a zero resultant is 3.
43. There are three laws of addition of vectors.
a) Commutative law:  
b) Associative law:    
c) Distributive law:      where m is a scalar.
44. If the number of vectors is more than two, polygon law of vectors is used.
45. Polygon law : If a number of vectors are represented by the sides of a polygon taken in the same order, the resultant is represented by the closing side of the polygon taken in the reverse order.



47. If  are unit vectors along X and Y axes, any vector lying in XOY plane can be represented as 
48.  ;  
49. The component of a vector can have a magnitude greater than that of the vector itself.
50. The rectangular component cannot have magnitude greater than that of the vector itself.

 

51. If a number of vectors  acting at a point are resolved along X – direction as Ax, Bx, Cx, Dx, … along Y– direction as Ay, By, Cy, Dy ….. and if   is the resultant of all the vectors, then the components of   along X – direction  and Y– direction are given by
Rx = Ax + Bx + Cx + Dx + … and Ry = Ay + By + Cy + Dy + … respectively, and 
 where θ is the angle made by the resultant with X – direction.

                                        
52.  If   are unit vectors along X, Y and Z–axes, any vector in 3 dimensional space can be expressed as

Here Ax, Ay, Az are the components of  and  are scalars.
 is body diagonal of the cube.

 

53. If α, β and γ are the angles made by  with X–axis, Y–axis and Z–axis respectively, then

cos2α + cos2β + cos2 γ = 1
sin2α + sin2 β + sin2 γ = 2


 

54. If cos α = l, cos β = m and cos γ = n, then l, m, n are called direction cosines of the vector.
l2 + m2 + n2 = 1.

 

55. If vectors  and 

and   = K   where K is a scalar.
 

56. The vector   is equally inclined to the coordinate axes at an angle of 54.74°.
 

57. The position vector of a point P(x, y, z) is given by


     

 

58. The vector having initial point P(x1, y1, z1) and final point Q(x2, y2, z2) is given by


      

 

59. Equilibrium is the state of a body in which there is no acceleration i.e., net force acting on a body is zero.
 

60. The forces whose lines of action pass through a common point (called the point of concurrence) are called concurrent forces.
 

61. Resultant force is the single force which produces the same effect as a given system of forces acting simultaneously.
 

62. A force which when acting along with a given system of forces produces equilibrium is called the equilibrant.
 

63. Resultant and equilibrant have equal magnitude and opposite direction. They act along the same line and they are themselves in equilibrium.

64. Triangle law of forces : If a body is in equilibrium under the action of three coplanar forces, then these forces can be represented in magnitude as well as direction by the three sides of a triangle taken in order.   where p, q, r are sides of a triangle.   are coplanar vectors.

65. Lami’s theorem : When three coplanar forces   keep a body in equilibrium, then      
 

66. When a number of forces acting on a body keep it in equilibrium, then the algebraic sum of the components along the X–direction is equal to zero and the algebraic sum of the components along the Y–direction is also equal to zero. i.e., ΣFx = 0 and ΣFy = 0.
 

67. If   and A1 = A2 = A3 = … An, then the adjacent vectors are inclined to each other at an angle 
 

68. N forces each of magnitude F are acting on a point and angle between any two adjacent forces is θ, then resultant force Fresultant

69. BODY PULLED HORIZONTALLY :
i) A body is suspended by a string from a rigid support. It is pulled aside so that it makes an angle θ with the vertical by applying a horizontal force F. When the body is in equilibrium,
ii) Horizontal force, F = mgTanθ
iii) Tension in the string, 
             
iv) T = 
v)     

 

70. If a body simultaneously possesses two velocities , the resultant velocity is given by the following formulae.
a) If   are in the same direction, the resultant velocity will be  in the direction of  or  .
b) If   are in opposite direction, the magnitude of the resultant velocity will be  and acts in the direction of the greater velocity.

c) If u  are mutually perpendicular, the resultant velocity will be  making an angle tan–1(v/u) with the direction of  .
 

71. If  is the velocity of the flow of water in a river and   is the velocity of a boat (relative to still water), then the velocity of the boat w.r.t the ground is  .
i. If the boat is going down stream, the velocity of the boat relative to the ground, .
ii. If the boat is going upstream, the velocity of the boat relative to the ground, .
iii. If the boat is moving at right angles to the stream,  making an angle of tan−1(v1/v2) with the original direction of the motion.

 

72. MOTION OF A BOAT CROSSING THE RIVER IN SHORTEST TIME :
    If   are the velocities of a boat and river flow respectively then to cross the river in shortest time, the boat is to be rowed across the river i.e., along normal to the banks of the river.


                                                      

i) The direction of the resultant is θ = tan–1  with the normal or tan θ = 
ii) Magnitude of the resultant velocity  
iii) Time taken to cross the river,  where

iv) This time is independent of velocity of the river flow.

v) The distance travelled down stream = BC


    

 

73. MOTION OF A BOAT CROSSING THE RIVER IN SHORTEST DISTANCE :
i) The boat is to be rowed upstream making some angle θ with normal to the bank of the river which is given by


                                                                    

ii) The angle made by boat with the bank or river current is (90° + θ)
iii) Resultant velocity has a magnitude of  V = 
iv) The time taken to cross the river is t = 

 

Subtraction of two vectors :
74.
If  are two vectors, then  –  is defined as  + (-) where -   is the negative vector of .
If   , then

In the parallelogram OMLN, the diagonal OL represents  and the diagonal NM represents    
a) subtraction of vectors does not obey commutative law 
b) subtraction of vectors does not obey Associative law  
c) subtraction of vectors obeys distributive law m 

 

75. If two vectors each of magnitude ‘F’ act at a point, the magnitude of their difference depends on the angle ‘θ’ between then
Magnitude of difference of vectors = 2Fsin 

 

76. Relative velocity : When the distance between two bodies is altering either in magnitude or direction or both, then each is said to have a relative velocity with respect to the other.
Relative velocity is vector difference of velocities.
a. The relative velocity of body 'A' w.r.t. 'B' is given by 
b. The relative velocity of body 'B' w.r.t. 'A' is given by 
 
c.  are equal in magnitude but opposite in direction
d. 
e. For two bodies moving in the same direction, relative velocity is equal to the difference of velocities. (θ = 0°.cos 0 = 1)
 = VA – VB

f. For two bodies moving in opposite direction, relative velocity is equal to the sum of their velocities. (θ =180°; cos180 = –1)
    = VA + VB
g. If they move at right angle to each other, then the relative velocity = .

77. Rain is falling vertically downwards with a velocity  and a person is travelling with a velocity  Then the relative velocity of rain with respect to the person is .
Relative velocity = .


78. The direction of relative velocity (or) the angle with the vertical at which an umbrella 

 

79. If the product of two vectors is another vector, such a product is called vector product or cross product.
 

80. If the product of two vectors is a scalar, then such a product is called scalar product or dot product.

 

DOT PRODUCT :
81.
Dot product is the product of one vector and the component of another vector in its direction.
Eg: Magnetic flux, instantaneous power, work done, potential energy.

 

82. Dot product of two vectors  and   = .
 = || || cos θ = AB cos θ
a) Scalar product is commutative i.e.,  
b) Scalar product is distributive i.e., 

 

83. If , and  are parallel vectors, then  . = AB.
 

84. If and  are perpendicular to each other, then .  = 0.
 

85. If  and   are antiparallel vectors, then  .  = –AB.
 

86. Dot product of two vectors may be positive or negative. If θ < 90°, it is positive and 90° < θ < 270° it is negative.
 

87. In the case of unit vectors, 
 

88. If  and  then  ,  

 

APPLICATIONS OF DOT PRODUCT :
89.
W =   (dot product of force and displacement is work)

 

90. P =  (dot product of force and velocity is power)
 

91. Ep =  (dot product of gravitational force and vertical displacement is P.E)
 

92. Magnetic flux, φ =   (dot product of area vector and magnetic flux density vector)
93. Angle between the two vectors  is given by Cosθ = 
94. The magnitude of component of vector  along vector .
95. The magnitude of component of vector   along vector .
96. Component of vector  along vector  =  .

 

CROSS PRODUCT :
97
. The vector or cross product of two vectors   is a vector  whose magnitude is AB sin q where q is the angle between the vectors  and the direction of  is perpendicular to both  such that  form a right hand triple.

Eg: angular momentum (), torque (), angular velocity () etc.
In the case of unit vectors


                

 

98. If  then


             

 

99. If   = 0 and   are not null vectors, then they are parallel to each other.


         
a)  (commutative law is not obeyed)
b)  (Associate law is not obeyed)
c)  (Distributive law is obeyed)

 

APPLICATIONS OF CROSS PRODUCT :
100.
Torque is the cross product of radius vector and force vector, 

 

101. Angular momentum is the cross product of radius vector and linear momentum, 
 

102. Linear velocity in circular motion may be defined as the cross product of angular velocity and radius vector. 
 

103. The area of the triangle formed by 


            

105. The area of the parallelogram formed by as adjacent sides is 
106. If  are diagonals of a parallelogram, then area of parallelogram =  

107.  Unit vector parallel to   or normal to  and 
        
108.  If  , then  are coplanar.

 

109.  If  , then .
 

110.  Division by a vector: is not defined because it is not possible to divide a direction by a direction.

Posted Date : 17-02-2021

గమనిక : ప్రతిభ.ఈనాడు.నెట్లో వచ్చే ప్రకటనలు అనేక దేశాల నుండి, వ్యాపారస్తులు లేదా వ్యక్తుల నుండి వివిధ పద్ధతులలో సేకరించబడతాయి. ఆయా ప్రకటనకర్తల ఉత్పత్తులు లేదా సేవల గురించి ఈనాడు యాజమాన్యానికీ, ఉద్యోగస్తులకూ ఎటువంటి అవగాహనా ఉండదు. కొన్ని ప్రకటనలు పాఠకుల అభిరుచిననుసరించి కృత్రిమ మేధస్సు సాంకేతికతతో పంపబడతాయి. ఏ ప్రకటనని అయినా పాఠకులు తగినంత జాగ్రత్త వహించి, ఉత్పత్తులు లేదా సేవల గురించి తగిన విచారణ చేసి, తగిన జాగ్రత్తలు తీసుకొని కొనుగోలు చేయాలి. ఉత్పత్తులు / సేవలపై ఈనాడు యాజమాన్యానికి ఎటువంటి నియంత్రణ ఉండదు. కనుక ఉత్పత్తులు లేదా సేవల నాణ్యత లేదా లోపాల విషయంలో ఈనాడు యాజమాన్యం ఎటువంటి బాధ్యత వహించదు. ఈ విషయంలో ఎటువంటి ఉత్తర ప్రత్యుత్తరాలకీ తావు లేదు. ఫిర్యాదులు తీసుకోబడవు.

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