ELEMENTS OF VECTOS 1 Scalar : physical quantity having only magnitue but not associate with any irection is calle a scalar eg: time, mass, istance, spee, work, energy, power, pressure, temperature, electric current, gravitational potential, pole strength, magnetic flu, entropy, electric capacity, velocity of light, large angular isplacement, electric charge, etc Scalars are ae an subtracte by algebraic metho 3 Vector : physical quantity having magnitue as well as associate irection an which obeys vector laws is calle a vector eg: isplacement, velocity, acceleration, force, momentum, impulse, moment of force, small angular isplacement, angular velocity, angular acceleration, magnetic moment, ipole moment, current ensity, intensity of electric fiel or magnetic fiel, shearing stress, weight, centrifugal force, infinitesimally small area, etc 4 Vectors are completely escribe by a number with a unit followe by a statement of irection 5 ngle can be consiere as vector if it is small Large angles can not be treate as vectors as they o not obey laws of vector aition 6 Surface area can be treate both as a scalar an a vector is magnitue of surface ˆn area which is a scalar This area is enclose by a close curve as shown if ˆn is a unit vector normal to the surface, we can write n ˆ as a vector Surface area is a vector [If the four fingers of right han curl along the irection of arrow of enclosing curve, thumb inicates irection of area vector] 7 Tensor is a physical quantity which will have ifferent values along ifferent irections eg Moment of inertia, stress 8 vector is represente by a irecte line segment The length of the line segment is proportional to the magnitue of the vector 9 The magnitue or moulus of a vector ( r or r) is a scalar 10 Electric current, velocity of light has both magnitue an irection but they o not obey the laws of vector aition Hence they are scalars 11 Equal vectors : Two vectors are sai to be equal if they have the same magnitue an irection irrespective of their initial points 1 Negative vectors : an are vectors having the same magnitue an opposite irection is calle the negative of 13 Proper vector : vector whose magnitue is not zero is known as proper vector 14 Null Vector (Zero Vector): It is a vector whose magnitue is zero an irection is unspecifie Eamples: a) Displacement after one complete revolution b) Velocity of vertically projecte boy at the highest point 15 Parallel vectors : Vectors in the same irection are calle parallel vectors 16 ntiparallel vectors : Vectors in opposite irection are calle antiparallel vectors 17 Like vectors or co-irectional vectors : The vectors irecte in the same irection, irrespective of their magnitues are calle co-irectional vectors or like vectors 3
18 Collinear vectors : Two or more vectors parallel or antiparallel to each other are calle collinear vectors 19 Coplanar vectors : Vectors lying on the same plane are calle coplanar vectors an the plane in which they lie is calle the plane of the vectors 0 Unit vector : It is a vector whose magnitue is unity unit vector parallel to a given vector 1 If is a vector, the unit vector in the irection of vector is written as a or  = = moulus of î, ĵ ankˆ are units vectors along, y an z ais Position vector : The vector which is use to specify the position of a point P with respect to some fie point O is represente by OP an is known P as the position vector of P with respect to O O 3 eal Vector or Polar Vector: If the irection of a vector is inepenent of the coorinate system, then it is calle a polar vector Eample: linear velocity, linear momentum, force, etc, 4 Pseuo or aial vectors : ial vectors or pseuo vectors are those whose irection is fie by convention an reverses in a mirror reflection Cross prouct of two vectors gives an aial vector eg : Torque, angular velocity, etc 5 vector remains unchange when it is move parallel to itself 6 If m is a scalar an a vector, then m is a vector Its magnitue is m times that of magnitue of Its irection is the same as that of, if m is positive an opposite if m is negative, P = mv ; F = ma ; F = Eq; F = m If m is zero, m is a null vector 7 Vector multiplication obeys commutative law when multiplie by a scalar s = s where s is scalar 8 Vector multiplication obeys associative law when multiplier by a scalar ie m(n )=mn (m, n are scalars) 9 Vector multiplication obeys istributive law when multiplie by a scalar s( + )=s +s DDITION OF VECTOS: 30 ition of vectors is also calle resultant of vectors 31 esultant is a single vector that gives the total effect of number of vectors esultant can be foun by using a) Triangle law of vectors b) Parallelogram law of vectors c) Polygon law of vectors 3 Two vectors can be ae either by triangle law or parallelogram law of vectors = P + Q 33 Triangle law : If two vectors are represente in magnitue an irection by the two sies of a triangle taken in orer, then the thir sie taken in the reverse orer represents their sum or resultant in magnitue as well as in irection P Q 34 Parallelogram law : If two vectors P an Q are represente by the two sies of a parallelogram rawn from a point, then their resultant is represente in magnitue an irection by Q β the iagonal of the parallelogram passing through that point α = P + Q + PQ cos P 4
Q sin P sin Tan α = ; tan β = P + Q cos Q + P cos 35 The resultant of two vectors is the vectorial aition of two vectors 36 The resultant of any two vectors makes lesser angle with the greater vector α 37 If > α < β β 38 The magnitue of the resultant of two vectors of magnitues a an b with arbitrary irections must be in the range (a b) to (a + b) 39 a an b are two vectors which when ae give a vector c (ie, ) a + b = c an if i) a + b = c then a an b are parallel vectors ( = 0 ) ii) a + b = c then a an b are perpenicular vectors ( = 90 ) iii) a b = c then a an b are antiparallel vectors ( = 180 ) iv) a = b = c then a an b are incline to each other at 10 v) If = an + =, then = 10 vi) If = an =, then = 60 vii) If + = -, then = 90 40 If two vectors each of magnitue F act at a point, the magnitue of their resultant () epens on the angle between them = Fcos( /) ngle between forces ( ) Magnitue of resultant 0 F 60 3 F 90 F 10 F 180 0 41 Minimum number of equal vectors to give a zero resultant is 4 The minimum number of unequal vectors to give a zero resultant is 3 43 There are three laws of aition of vectors a) Commutative law: + = + b) ssociative law: + ( + C) = ( + ) + C c) Distributive law: m( + ) = m + m where m is a scalar 44 If the number of vectors is more than two, polygon law of vectors is use D C 45 Polygon law : If a number of vectors are represente by the sies of a polygon taken in the same orer, the resultant is represente by the closing sie of the polygon taken in the reverse orer = + + C + D 5
46 esolution of a vector in two imensions : If is a vector making an angle with ais, then X component = cos, Y component = sin 47 If î an ĵ are unit vectors along X an Y aes, any vector lying in XOY plane can be represente as = î ĵ ; + y 6 Elements of Vectors y 48 = = + y ; Tan = 49 The component of a vector can have a magnitue greater than that of the vector itself 50 The rectangular component cannot have magnitue greater than that of the vector itself 51 If a number of vectors,, C,D, acting at a point are resolve along Y X irection as,, C, D, along Y irection as y, y, C y, D y an if is the resultant of all the vectors, then the components of y ĵ along X irection an Y irection are given by = + + C + D + an y = y + y + î X C y + D y + respectively, an z kˆ y = Z + y ; tan = where is the angle mae by the resultant with X irection 5 If î, ĵ an kˆ are unit vectors along X, Y an Z aes, any vector in 3 imensional space can be epresse as = î + y ĵ + z kˆ ; = = + y + z Here, y, z are the components of an are scalars is boy iagonal of the cube 53 If α, β an γ are the angles mae by with X ais, Y ais an Z ais respectively, then y z cos α = ; cosβ = ; cosγ = an P(,y,z) cos α + cos β + cos γ = 1 sin α + sin β + sin γ = 54 If cos α = l, cos β = m an cos γ = n, then l, m, n are calle irection cosines of the vector l + m + n = 1 55 If vectors = î + y ĵ + zkˆ an ˆ ˆ ˆ y z = i + yj+ zk are parallel, then = = an =K y z where K is a scalar 56 The vector î + ĵ + kˆ is equally incline to the coorinate aes at an angle of 5474 57 The position vector of a point P(,y,z) is given by OP = î + yĵ + zkˆ an OP = + y + z 58 The vector having initial point P( 1, y 1, z 1 ) an final point Q(, y, z ) is given by PQ = ( 1)î + (y y1)ĵ + (z z1)kˆ 59 Equilibrium is the state of a boy in which there is no acceleration ie, net force acting on a boy is zero 60 The forces whose lines of action pass through a common point (calle the point of concurrence) are calle concurrent forces Y O Y O O (origin) cos y j X i sin X
61 esultant force is the single force which prouces the same effect as a given system of forces acting simultaneously 6 force which when acting along with a given system of forces prouces equilibrium is calle the equilibrant 63 esultant an equilibrant have equal magnitue an opposite irection They act along the same line an they are themselves in equilibrium 64 Triangle law of forces : If a boy is in equilibrium uner the action of three coplanar forces, then these forces can be represente in magnitue as well as r P p p q r q irection by the three sies of a triangle taken in orer = = where p, q, P Q Q r are sies of a triangle P, Q, are coplanar vectors P Q 65 Lami s theorem : When three coplanar forces P, Q an keep a boy in γ P Q β α equilibrium, then = = sinα sinβ sin γ 66 When a number of forces acting on a boy keep it in equilibrium, then the algebraic sum of the components along the X irection is equal to zero an the algebraic sum of the components along the Y irection is also equal to zero ie, F = 0 an Fy = 0 67 If 1 + + 3 + + n = 0 an 1 = = 3 = n, then the ajacent vectors are incline to each π 360 other at an angle or N N 68 N forces each of magnitue F are acting on a point an angle between any two ajacent forces is, N Fsin then resultant force F resultant = sin( / ) 69 ODY PULLED HOIZONTLLY : i) boy is suspene by a string from a rigi support It is pulle asie so that it makes an angle with the vertical by applying a horizontal force F When the boy is in equilibrium, ii) Horizontal force, F= mgtan l iii) Tension in the string, mg T T = F Cos iv) T= ( mg) + F T mg F mg v) = = l l 70 If a boy simultaneously possesses two velocities u an v, the resultant velocity is given by the following formulae a) If u an v are in the same irection, the resultant velocity will be u + v in the irection of u or v b) If u an v are in opposite irection, the magnitue of the resultant velocity will be u ~ v an acts in the irection of the greater velocity c) If u an v are mutually perpenicular, the resultant velocity will be u + v making an angle tan 1 (v/u) with the irection of u l - 7
71 If v 1 is the velocity of the flow of water in a river an v is the velocity of a boat (relative to still water), then the velocity of the boat wrt the groun is v + 1 v i If the boat is going own stream, the velocity of the boat relative to the groun, v = v1 + v ii If the boat is going upstream, the velocity of the boat relative to the groun, v = v1 v iii If the boat is moving at right angles to the stream, v = v 1 + v making an angle of tan 1 (v 1 /v ) with the original irection of the motion 7 MOTION OF OT COSSING THE IVE IN SHOTEST TIME : If V an V are the velocities of a boat an river flow respectively then to C cross the river in shortest time, the boat is to be rowe across the river ie, along normal to the banks of the river V V i) The irection of the resultant is = tan 1 V with the normal or tan V V = = V ii) Magnitue of the resultant velocity v = v + v iii) Time taken to cross the river, t = =with of the river or t= v where + = V V + V iv) This time is inepenent of velocity of the river flow v) The istance travelle own stream = C = V V 73 MOTION OF OT COSSING THE IVE IN SHOTEST DISTNCE : i) The boat is to be rowe upstream making some angle with normal to the bank of the river which is given by = sin 1 V V C or sin= V V ii) The angle mae by boat with the bank or river current is (90 + ) iii) esultant velocity has a magnitue of V = V V iv) The time taken to cross the river is t = 8 V V Subtraction of two vectors : 74 IfP an Q are two vectors, then P Q is efine as P + ( Q) where Q is the negative vector of Q If = P Q, then = P + Q PQCos In the parallelogram OMLN, the iagonal OL represents + N an the iagonal L NM represents a) subtraction of vectors oes not obey commutative law O M b) subtraction of vectors oes not obey ssociative law ( C) ( ) C V V
c) subtraction of vectors obeys istributive law m( ) = m m 75 If two vectors each of magnitue F act at a point, the magnitue of their ifference epens on the angle between then Magnitue of ifference of vectors = Fsin 76 elative velocity : When the istance between two boies is altering either in magnitue or irection or both, then each is sai to have a relative velocity with respect to the other elative velocity is vector ifference of velocities a The relative velocity of boy '' wrt '' is given by V = V V b The relative velocity of boy '' wrt '' is given by V = V V c V V an V V are equal in magnitue but opposite in irection V = V V = V + V V V cos e For two boies moving in the same irection, relative velocity is equal to the ifference of velocities ( = 0 cos 0 = 1) V = V V f For two boies moving in opposite irection, relative velocity is equal to the sum of their velocities ( =180 ;cos180 = 1) V =V + V g If they move at right angle to each other, then the relative velocity -V P V P = v 1 + v 77 ain is falling vertically ownwars with a velocity V V an a person is V travelling with a velocity V P Then the relative velocity of rain with respect α to the person is V = V VP elative velocity = V = V + VP 78 The irection of relative velocity (or) the angle with the vertical at which an umbrella is to be hel VP is given by Tan = V 79 If the prouct of two vectors is another vector, such a prouct is calle vector prouct or cross prouct 80 If the prouct of two vectors is a scalar, then such a prouct is calle scalar prouct or ot prouct DOT PODUCT : 81 Dot prouct is the prouct of one vector an the component of another vector in its irection Eg : Magnetic flu, instantaneous power, work one, potential energy 8 Dot prouct of two vectors an = = cos = cos a) Scalar prouct is commutative ie, ab = b a b) Scalar prouct is istributive ie, a(b + c) = ab + a c 83 If an are parallel vectors, then = 9
84 If an are perpenicular to each other, then =0 85 If an are antiparallel vectors, then = 86 Dot prouct of two vectors may be positive or negative If <90, it is positive an 90 <<70 it is negative 87 In the case of unit vectors, î î = ĵ ĵ = kˆ kˆ = 1 an î ĵ = ĵ kˆ = kˆ î = 0 88 If ˆ ˆ ˆ = i + yj+ zk an ˆ ˆ ˆ = i + yj + zk, then = + yy + zz, = + y + z PPLICTIONS OF DOT PODUCT : 89 W = F S (ot prouct of force an isplacement is work) 90 P = F V (ot prouct of force an velocity is power) 91 E p = mg h (ot prouct of gravitational force an vertical isplacement is PE) 9 Magnetic flu, φ = (ot prouct of area vector an magnetic flu ensity vector) b 93 ngle between the two vectors a an b is given by Cos = a a b 94 The magnitue of component of vector along vector = 95 The magnitue of component of vector along vector = 96 Component of vector along vector = ˆ COSS PODUCT : 97 The vector or cross prouct of two vectors an is a vector C whose magnitue is sin where is the angle between the vectors an an the irection of C is perpenicular to both an such that, an C form a right han triple Eg: angular momentum ( L = r ω ), torque ( τ = r F ), angular velocity ( V = ω r ) etc î ĵ = kˆ ; ĵ kˆ = î ; kˆ î = ĵ ; In the case of unit vectors ĵ î = - kˆ;kˆ ĵ = - î; î kˆ = - ĵ an î î = ĵ ĵ = kˆ kˆ = 0 98 If = î + ĵ + kˆ an = î + ĵ kˆ, then î = y z y + ( yz zy )î (z z )ĵ + (y y )kˆ = 0 an an = ( + C) = + C ; m( ) = (m) = (m) 99 If a) (commutative law is not obeye) b) ( C) ( ) C z are not null vectors, then they are parallel to each other 30 ĵ y y kˆ z z =
(ssociate law is not obeye) c) ( + C) = + C (Distributive law is obeye) PPLICTIONS OF COSS PODUCT : 100 Torque is the cross prouct of raius vector an force vector, τ = r F 101 ngular momentum is the cross prouct of raius vector an linear momentum, L = r p 10 Linear velocity in circular motion may be efine as the cross prouct of angular velocity an raius vector V = ω r 1 103 The area of the triangle forme by an as ajacent sies is 104 rea of triangle C if position vector of is a, position vector of is b an position vector of C is c, then area = 1 a b + b c + c a 105 The area of the parallelogram forme by an as ajacent sies is 106 If P an Q are iagonals of a parallelogram, then area of parallelogram= 1 (P Q ) 107 Unit vector parallel to C or normal to an is n = 108 If ( C) = 0, then, an C are coplanar 109 If + = C, then ( C) = 0 110 Division by a vector: is not efine because it is not possible to ivie a irection by a irection 31