fixed point ( The fixed point is also called as origin) and P is any point then OP is the position vector of the point P with respect the point O.

Transcription

1 Page 1 of 9 VECTORS 1 Scala Quantity: A scala quantity is that which has only magnitude Example: Volume, Aea, Tempeatue, wok done, time, density etc ae scala quantities as these quantities have no sense of diection Vecto Quantity: Quantity having magnitude and diection both is called as vecto Example: Foce, Velocity, Acceleation, Weight, Momentum, Displacement, electic field intensity etc ae vecto quantities 3 Diected line segment: A line segment is called as diected line segment if its initial point (stating point) and teminal point (initial point) ae specified ie If a line segment has initial point A and teminal point B then it is a diected line segment AB denoted as AB o simply by using small English alphabet such as a and is epesented as follows Length of the vecto epesents the magnitude of the vecto and is denoted as AB o a 4 Types of vectos (i) Position vecto: A vecto that epesents the position of a point with espect to given fixed point is called as position vecto of the given point ie if O is the fixed point ( The fixed point is also called as oigin) and P is any point then OP is the position vecto of the point P with espect the point O Note: A position vecto fixes the position of a point 5 Zeo Vecto: A vecto having zeo magnitude is called a zeo vecto Note: A zeo vecto has no paticula diection o it may be consideed to have any diection Example: AA is a zeo vecto Hee AA is nothing but the point A 6 Unit Vecto: A vecto having magnitude 1 is called as unit vecto Note: Unit vecto along the diection of a vecto: If a is a vecto then a unit vecto along it is given by a and is denoted by â ie a ˆ a a = a 7 Coinitial Vectos: Vectos having same initial point ae called as Coinitial Vectos Example: AB, AC, AD AE etc ae Coinitial Vectos 8 Collinea Vectos: Vectos which ae paallel to each othe iespective of thei magnitude and diections ae called as Collinea Vectos Example: The vectos AB, CD, and EF as shown in the figue ae collinea vectos 9 Negative of a vecto: Negative of a vecto is a vecto which has magnitude as that of the given vecto but diection opposite to the given vecto Example: If AB is a vecto then BA is it s negative vecto ie BA = AB o AB = BA Pepaed and designed by Ajay Mawaha Please tun ove

2 Page of 9 10 Equal vectos: Two o moe vectos ae said to be equal vectos iff they have same diection and same magnitude Example: In the figue AB = CD = EF Note: Equal vectos ae always collinea vectos 11 Fee Vectos: A vecto that can be displaced paallel to itself is called as fee vecto Note: We will conside the vectos in ou syllabus as fee vectos unless othewise stated ADDITION OF VECTORS Vectos can be added by two methods: METHOD1 Tiangle law of vecto addition: Addition of two vectos that can be epesented in magnitude and diection by the two sides of a tiangle in the same ode is the vecto epesented in magnitude and diection by the thid side of the tiangle in the evese ode Example: In the figue: AB + BC = AC Note: 1 Sum of the thee vecto that can be epesented in magnitude and diection by the thee sides of a tiangle in the same ode ( diection) is a zeo vecto ie AB + BC + CA = 0 If AB is any vecto then AB = OB OA = Position vecto of 'B' Postion vecto of 'A' As shown in the following figue: METHOD Paallelogam law of vecto addition: If two vectos can be epesented in magnitude and diection by the two adjacent sides of a paallelogam then thei addition is a vecto that can be epesented in magnitude and diection by the diagonal of the paallelogam Coinitial with the given vectos Example: In the figue: AB + AD = AC NOTE: In the above figue DB = AB AD and BD = AD AB Pepaed and designed by Ajay Mawaha Please tun ove

3 Page 3 of 9 Popeties of vecto additions 1 Vecto addition is commutative: ie a + b = b + a Vecto addition is associative: a + b + c = a + b + c ie ( ) ( ) 3 Zeo vecto is the additive identity ie 0 + a = a + 0 SCALAR MULTIPLICATION OF VECTOR Scala multiplication of a vecto is a vecto that is obtained by multiplying the vecto by a scala ie If a is any vecto and λ is a scala ie λ R then scala multiplication of the vecto a with the scala λ is the vecto λ a, whee (i) λ a = λ a (ii) If λ > 0 then a and λ a have same diection If λ < 0 then a and λ a have opposite diection Example: a, 1 a, a 1, a, a ae shown in the following figue: POSITION VECTOR OF A POINT IN CARTESIAN FORM: (I) In two dimension ( ie in D): If P(x,y) is a point on plane then OP (the position vecto of P) with espect to system of axis will be given by OP = xiˆ + yj ˆ as shown in the figue below: FIGURE ON NEXT PAGE Pepaed and designed by Ajay Mawaha Please tun ove

4 Page 4 of 9 OP = xiˆ + yj ˆ P(x, y) ˆ yj O ˆ xi A x-axis y-axis NOTE: OP = x + y (II) In thee dimension ( ie in 3D): If P(x,y,z) is a point on plane then OP (the position vecto of P) with espect to system of axis in the space will be given by OP = xiˆ + yj ˆ + zkˆ as shown in the figue below: In the above figue OP = OQ + QP ˆ ˆ ˆ 1 + P1 P = xi + yj + zk NOTE: OP = x + y + z RESULTS:, 1 If A ( x y ) and (, ) 1 1 B x y then OA = x ˆ ˆ 1i + y1 j and OB = x ˆ ˆ i + y j AB = OB OA = x x iˆ + y y ˆj Theefoe, ( ) ( ) NOTE: AB = ( x x ) + ( y y ) Pepaed and designed by Ajay Mawaha Please tun ove

5 Page 5 of 9 If A ( x, y, z ) and (,, ) B x y z then ˆ OA = x ˆ ˆ 1i + y1 j + z1k and OB = x ˆ ˆ ˆ i + y j + zk AB = OB OA = x ˆ ˆ ˆ x1 i + y y1 j + z z1 k AB = x x + y y + z z Theefoe, ( ) ( ) ( ) NOTE: ( ) ( ) ( ) ANGLE BETWEEN TWO VECTORS: Angle between two vectos is the angle between the Coinitial vectos displaced paallel to the given vectos ie If AB and CD ae two vectos then the angle between them is the angle PQR between the coinitial vectos QP and QR such that QP = AB and QR = CD DIRECTION COSINES AND DIRECTION RATIOS OF A VECTOR Let AB be any vecto in the space and OP be the vecto paallel to AB then angle between AB and x-axis, y-axis and z-axis is same as the angle between the vecto OP and x-axis, y-axis and z-axis espectively If ( α, β, γ ) ae the angles between the OP and x-axis, y-axis and z-axis espectively then diection cosines of the vecto AB ae ( cos α,cos β,cos γ ) espectively, also denoted as ( l, m, n) ie cos α = l, cos β = m and cos γ =n as shown in the figue below: NOTE: l + m + n = 1ie sum of the squaes of the diection cosines of a vecto is always one DIRECTION RATIOS OF A VECTOR: Diection atio of a vecto is the set of thee numbes which ae popotional to the diection cosines of a given vecto,, a, b, c is the set of ie if ( l m n ) ae the dc s (diection cosines) of a vecto and ( ) thee numbes such that l m n a b c (diection atios) of the given vecto = = then we say that (,, ) a b c ae the d-atio s Pepaed and designed by Ajay Mawaha Please tun ove

6 Page 6 of 9 RESULTS: 1 If ˆ a = a ˆ ˆ 1i + a j + a3k and b = b ˆ ˆ ˆ 1i + b j + b3k then a = b iff a1 = b1, a = b, and a3 = b3 If a = a ˆ ˆ ˆ 1i + a j + a3k then ( a1, a, a3 ) ae the diection atios of a and its d- cosines ae given by: a1 a a3 l =, m = and n = a a a 3 Whee, a = a1 + a + a3 3 If a = a ˆ ˆ ˆ 1i + a j + a3k then a1ˆ i, a ˆ j and a ˆ 3k ae called as the components of a 4 Popeties of scala multiplication: If k and m ae any two eal numbes then (i) ka + ma = ( k + m) a (ii) k ( ma ) = ( km ) a k a + b = ka + kb (iii) ( ) l m n ae the d-cosines of a vecto a then a unit vecto â along a is given by â liˆ mj ˆ zkˆ a = a liˆ + mj ˆ + nkˆ 5 If (,, ) = + + also ( ) SECTION FORMULAE: If P and Q ae two points with position vectos a and b then position vecto of a point R which CASE I: Divides the vecto PQ intenally in the atio of m : n is given by mb + na OR = m + n NOTE: Position vecto of mid point R of PQ, whee P and Q ae two points with position vectos a and b a + b is given by OR = CASE II: Divides the vecto PQ extenally in the atio of m : n is given by mb na OR = m n Pepaed and designed by Ajay Mawaha Please tun ove

7 Page 7 of 9 DOT PRODUCT (SCALAR PRODUCT OF TWO VECTORS) Scala poduct of two vectos a and b is defined as a b = a b cosθ Whee, θ is angle between a andb with 0 θ π RESULTS: 1 a b is a eal numbe Let a and b be two nonzeo vectos, then a b = 0 if and only if a and b ae pependicula to each othe ie If a 0 andb 0 then a b = 0 a b 3 If θ = 0,then a b = a b NOTE: a = a a = a,as θ in this case is 0 4 If θ = π then a b = a b a a = a as in this case θ = π NOTE: ( ) 5 As iˆ, ˆj & kˆ ae unit vectos ae pependicula to each othe theefoe iˆ iˆ = ˆj ˆj = kˆ kˆ = 1 & iˆ ˆj = ˆj kˆ = kˆ iˆ = 0 6 The angle θ between two nonzeo vectos a and b is given by a b cosθ = 1 a b,o θ = cos a b a b Popeties of scala poduct 1 The scala poduct is commutative ie a b = b a The scala poduct is associative ie a b + c = a b + a c ( ) 3 Let a and b be any two vectos, and λ be any scala Then λ a b = λ a b = a λ b ( ) ( ) ( ) Pepaed and designed by Ajay Mawaha Please tun ove

8 Page 8 of 9 PROJECTION OF A VECTOR ON A LINE / VECTOR 1 Pojection of a vecto on a line: Pojection p of a vecto a on a line l is given by: p = a cosθ o p = AB cosθ Pojection of a o AB on a line is the intecept of the pependiculas to the line though the end points of the vecto AB The pojections of vecto AB in diffeent diections ae shown as follows: RESULTS: 1 If ˆp is the unit vecto along a line l, then the pojection of a vecto a on the line l is given by a pˆ Pojection of a vecto a on othe vectob, is given by b 1 aˆ b o a, o ( a b ) b b 3 Ifθ = 0, then the pojection vecto of AB will be AB itself and if,θ = π then the pojection vecto of AB will be BA π 3π 4 If θ = oθ =, then the pojection vecto of AB will be zeo vecto VECTOR (OR CROSS) PRODUCT OF TWO VECTORS The vecto poduct of two nonzeo vectos a andb, is denoted by a b and defined as a b = a b sinθ nˆ Whee θ is the angle between a and b,0 θ π and ˆn is a unit vecto pependicula to both a and b, such that a,b and ˆn fom a ight handed system ie the ight handed system otated fom a tob moves in the diection of ˆn Pepaed and designed by Ajay Mawaha Please tun ove

9 Page 9 of 9 RESULTS 1 a b is avcto a = 0 ob = 0 then a b = 0 3 Let a and b be two nonzeo vectos Then a b = 0 if and only if a and b ae paallel (o collinea) to each othe ie, If a 0 and b 0 then a b = 0 a P b 4 a a = 0 and a ( a ) = 0, since in the fist situation, θ = 0 and in the second one, θ = π, making the value of sinθ to be 0 π 5 If θ = then a b = a b 6 Fo the unit vectos iˆ, ˆj and ˆk, we have iˆ iˆ = ˆj ˆj = kˆ kˆ = 0 iˆ ˆj = kˆ, ˆj kˆ = iˆ, kˆ iˆ = ˆj 7 In tems of vecto poduct, the angle between two vectos a and b may be given a b a b 1 as sinθ = o θ = sin a b a b 8 Vecto poduct is not commutative, infact a b = b a 9 If a and b epesent the adjacent sides of a tiangle then its aea is given as 1 a b 10 If a and b epesent the adjacent sides of a paallelogam, then its aea is given as a b 11 If ˆ a = a ˆ ˆ 1i + a j + a3k andb = b ˆ ˆ ˆ 1i + b j + b3k then î j kˆ a b = a a a Popeties: 1 a ( b + c ) = a b + a c λ a b = λ a b = a λ b ( ) ( ) ( ) 1 3 b b b 1 3 3, whee λ is any scala Pepaed and designed by Ajay Mawaha Please tun ove