BASIC ALGEBRA OF VECTORS

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1 Fomulae Fo u Vecto Algeba By Mi Mohammed Abbas II PCMB 'A' Impotant Tems, Definitions & Fomulae 01 Vecto - Basic Intoduction: A quantity having magnitude as well as the diection is called vecto It is denoted as AB o a Its magnitude (o modulus) is AB o a othewise, simply AB o a Vectos ae denoted by symbols such as a o a o a [Pictoial epesentation of vecto] 0 Initial and Teminal points: The initial and teminal points mean that point fom which the vecto oiginates and teminates espectively 03 Position Vecto: The position vecto of a point say P( x, y, z ) is OP x i ˆ y ˆ j z k ˆ and the magnitude is x y z The vecto OP x i ˆ y ˆ j z k ˆ is said to be in its component fom Hee x, y, z ae called the scala components o ectangula components of and xiˆ, yˆj, zk ˆ ae the vecto components of along x-, y-, z- axes espectively Also, AB Position Vecto of B Position Vecto of A Fo example, let A( x1, y 1,z 1) and B( x, y,z ) Then, AB ( x ˆ ˆ ˆ ˆ ˆ ˆ i y j zk) ( x1i y1 j z1k ) Hee iˆ, ˆj and ˆk ae the unit vectos along the axes OX, OY and OZ espectively (The discussion about unit vectos is given late in the point 05(e)) 04 Diection atios and diection cosines: If xiˆ yˆ j zkˆ, then coefficients of iˆ, ˆj, k ˆ in ie, x, y, z ae called the diection atios (abbeviated as d s) of vecto These ae denoted by a, b, c (ie a x, b y, c z ; in a manne we can say that scala components of vecto and its d s both ae the same) Also, the coefficients of iˆ, ˆj, k ˆ in ˆ (which is the unit vecto of x y ) ie,,, x y z x y z z ae called diection cosines (which is abbeviated as dc s) of vecto x y z These diection cosines ae denoted by l, m, n such that l cos, m cos, n cos and l m n 1 cos cos cos 1 x y z It can be easily concluded that l cos, m cos, n cos Theefoe, liˆ mˆj nkˆ (cos iˆ cos ˆj cos k ˆ) (Hee ) [See the OAP in Fig1] Angles,, ae made by the vecto with the positive diections of x, y, z-axes espectively and these angles ae known as the diection angles of vecto ) BASIC ALGEBRA OF VECTORS Fo a bette undestanding, you can visualize the Fig1 05 TYPES OF VECTORS a) Zeo o Null vecto: Its that vecto whose initial and teminal points ae coincident It is denoted by 0 Of couse its magnitude is 0 (zeo) Any non-zeo vecto is called a pope vecto b) Co-initial vectos: Those vectos (two o moe) having the same initial point ae called the coinitial vectos Page - [1] wwwmohammedabbas7wodpesscom

2 c) Co-teminous vectos: Those vectos (two o moe) having the same teminal point ae called the co-teminous vectos d) Negative of a vecto: The vecto which has the same magnitude as the but opposite diection It is denoted by Hence if, AB BA That is AB BA, PQ QP etc e) Unit vecto: It is a vecto with the unit magnitude The unit vecto in the diection of vecto is given by ˆ such that 1 So, if xiˆ yˆj zkˆ then its unit vecto is: x ˆ y ˆ z ˆ i j kˆ x y z x y z x y z Unit vecto pependicula to the plane of a a b and b is: a b f) Recipocal of a vecto: It is a vecto which has the same diection as the vecto but magnitude equal to the ecipocal of the magnitude of It is denoted as 1 Hence 1 1 g) Equal vectos: Two vectos ae said to be equal if they have the same magnitude as well as diection, egadless of the positions of thei initial points a b Thus a b a and b have same diection Also, if a b a1iˆ a ˆ ˆ ˆ ˆ ˆ j a3k b1i b j b3k a1 b1, a b, a3 b 3 h) Collinea o Paallel vecto: Two vectos a and b ae collinea o paallel if thee exists a nonzeo scala such that a b It is impotant to note that the espective coefficients of iˆ, ˆj, k ˆ in a and b ae popotional povide they ae paallel o collinea to each othe Conside ˆ ˆ ˆ ˆ ˆ ˆ a a1i a j a3k, b b1i b j b3k, then by using a b, we can conclude a1 a a3 that: b b b 1 3 The d s of paallel vectos ae same (o ae in popotion) The vectos a and b will have same o opposite diection as is positive o negative The vectos a and b ae collinea if a b 0 i) Fee vectos: The vectos which can undego paallel displacement without changing its magnitude and diection ae called fee vectos 06 ADDITION OF VECTORS a) Tiangula law: If two adjacent sides (say sides AB and BC) of a tiangle ABC ae epesented by a and b taken in same ode, then the thid side of the tiangle taken in the evese ode gives the sum of vectos a and b ie, AC AB BC AC a b See Fig Also since AC CA AB BC CA AA 0 And AB BC AC AB BC AC 0 AB BC CA 0 b) Paallelogam law: If two vectos a and b ae epesented in magnitude and the diection by the two adjacent sides (say AB and AD) of a paallelogam ABCD, then thei sum is given by that diagonal of paallelogam which is co-initial with a and b ie, OC OA OB Fo the illustation, see Fig3 Multiplication of a vecto by a scala Let a be any vecto and k be any scala Then the poduct ka is defined as a vecto whose magnitude is k times that of a and the diection is (i) same as that of a if k is positive, and (ii) opposite to that of a if k is negative Page - [] wwwmohammedabbas7wodpesscom

3 07 PROPERTIES OF VECTOR ADDITION Commutative popety: a b b a Conside ˆ ˆ ˆ a a1i a j a3k and b b ˆ ˆ ˆ 1i b j b3k be any two given vectos Then a b a1 b1 iˆ a b ˆj a3 b3 kˆ b a Associative popety: a b c a b c Additive identity popety: a 0 0 a a Additive invese popety: a ( a) 0 ( a) a 08 Section fomula: The position vecto of a point say P dividing a line segment joining the points A and B whose position vectos ae a and b espectively, in the atio m:n mb na (a) intenally, is OP m n mb na (b) extenally, is OP m n a b Also if point P is the mid-point of line segment AB then, OP Impotant Tems, Definitions & Fomulae 01 PRODUCT OF TWO VECTORS a) Scala poduct o Dot poduct: The dot poduct of two vectos a and b is defined by, a b a b cos whee θ is the angle between a and b, 0 See Fig4 Conside ˆ ˆ ˆ ˆ ˆ ˆ a a1i a j a3k, b b1i b j b3k Then a b a1b 1 ab a3b3 Popeties / Obsevations of Dot poduct iˆ iˆ iˆ iˆ cos0 1 iˆ iˆ 1 ˆj ˆj kˆ kˆ iˆ ˆj iˆ ˆj cos 0 iˆ ˆj 0 ˆj kˆ kˆ iˆ a b R, whee R is eal numbe ie, any scala a b b a (Commutative popety of dot poduct) a b 0 a b If θ = 0 then, a b a b Also a a a a ; as θ in this case is 0 Moeove if θ = then, a b a b a b c a b a c (Distibutive popety of dot poduct) PRODUCT OF VECTORS DOT PRODUCT & CROSS PRODUCT a b a b a b Angle between two vectos a and b can be found by the expession given below: cos a b a b Page - [3] wwwmohammedabbas7wodpesscom

4 1 o, cos a b a b Pojection of a vecto a on the othe vecto say b a b is given as ie, a bˆ b This is also known as Scala pojection o Component of a along b Pojection vecto of a on the othe vecto say b a b is given as b ˆ b This is also known as the Vecto pojection Wok done W in moving an object fom point A to the point B by applying a foce F is given as W F AB b) Vecto poduct o Coss poduct: The coss poduct of two vectos a and b is defined by, a b a b sin ˆ, whee is the angle between the vectos a and b, 0 and ˆ is a unit vecto pependicula to both a and b Fo bette illustation, see Fig5 Conside a a ˆ ˆ ˆ ˆ ˆ ˆ 1i a j a3k, b b1i b j b3k iˆ ˆj kˆ a b a a a a b a b iˆ a b a b ˆj a b a b kˆ Then, b b b 1 3 Popeties / Obsevations of Coss poduct iˆ iˆ iˆ iˆ sin 0 ˆj 0 iˆ iˆ 0 ˆj ˆj kˆ kˆ iˆ ˆj iˆ ˆj sin kˆ kˆ iˆ ˆj kˆ, ˆj kˆ iˆ, kˆ iˆ ˆj Fig6 at the end of chapte can be consideed fo memoizing the vecto poduct of iˆ, ˆj, k ˆ a b is a vecto c (say) and this vecto c is pependicula to both the vectos a and b a b a b sin ˆ a b sin ie, a b ab sin a b 0 a // b o, a 0, b 0 a a 0 a b b a (Commutative popety does not hold fo coss poduct) a b c a b a c; b c a b a c a (Distibutive popety of the vecto poduct o coss poduct) Angle between two vectos a and b in tems of Coss-poduct can be found by the ab expession given hee: sin a b a b 1 o, sin a b If a and b epesent the adjacent sides of a tiangle, then the aea of tiangle can be obtained by evaluating 1 a b Page - [4] wwwmohammedabbas7wodpesscom

5 If a and b epesent the adjacent sides of a paallelogam, then the aea of paallelogam can be obtained by evaluating a b If p and q epesent the two diagonals of a paallelogam, then the aea of paallelogam can be obtained by evaluating 1 p q If a and b epesent the adjacent sides of a paallelogam, then the diagonals d 1 and d of the paallelogam ae given as: d1 a b, d b a 0 Relationship between Vecto poduct and Scala poduct [Lagange s Identity] Conside two vectos a and b We also know that a b a b sin ˆ Now, a b a b sin ˆ a b a b sin a b a b sin a b a b 1 cos a b a b a b cos a b a b a b cos a b a b a b o, a b a b a b a a a b Note that a b Hee the RHS epesents a deteminant of ode a b b b 03 Cauchy- Schwatz inequality: Fo any two vectos a and b, we always have a b a b Poof: The given inequality holds tivially when eithe a 0 o b 0 ie, in such a case a b 0 a b So, let us check it fo a 0 b As we know, a b a b cos a b a b cos Also we know cos 1 fo all the values of a b cos a b a b a b a b a b [HP] 04 Tiangle inequality: Fo any two vectos a and b, we always have a b a b Poof: The given inequality holds tivially when eithe a 0 o b 0 ie, in such a case we have Page - [5] wwwmohammedabbas7wodpesscom

6 a b 0 a b So, let us check it fo a 0 b Then conside a b a b a b a b a b a b cos Fo cos 1, we have: a b cos a b a b a b cos a b a b a b a b a b a b [HP] SCALAR TRIPLE PRODUCT OF VECTORS Impotant Tems, Definitions & Fomulae 01 SCALAR TRIPLE PRODUCT: If a,b and c ae any thee vectos, then the scala poduct of poduct of a,b and c Thus, ( a b) c is called the scala tiple poduct of a,b and c a b with c is called scala tiple Notation fo scala tiple poduct: The scala tiple poduct of a,b and c is denoted by [ a b c ] That is, ( a b) c [ a b c ] Scala tiple poduct is also known as mixed poduct because in scala tiple poduct, both the signs of dot and coss ae used ˆ ˆ ˆ ˆ ˆ ˆ a a i a j a k, b b i b j b k, c c iˆ c ˆj c k ˆ Conside a a a Then, [ a b c] b b b c c c 1 3 Popeties / Obsevations of Scala Tiple Poduct ( a b) c a( b c ) That is, the position of dot and coss can be intechanged without change in the value of the scala tiple poduct (povided thei cyclic ode emains the same) [ a b c] [ b c a] [ c a b ] That is, the value of scala tiple poduct doesn t change when cyclic ode of the vectos is maintained Also [ a b c] [ b a c]; [ b c a] [ b a c ] That is, the value of scala tiple poduct emains the same in magnitude but changes the sign when cyclic ode of the vectos is alteed Fo any thee vectos a, b, c and scala, we have [ a b c] [ a b c ] The value of scala tiple poduct is zeo if any two of the thee vectos ae identical That is, [ a a c] 0 [ a b b] [ a b a ] etc Value of scala tiple poduct is zeo if any two of the thee vectos ae paallel o collinea Scala tiple poduct of iˆ, ˆj and ˆk is 1 (unity) ie, [ iˆ ˆj k ˆ] 1 Page - [6] wwwmohammedabbas7wodpesscom

7 If [ a b c ] 0 then, the non-paallel and non-zeo vectos a, b and c ae coplana Volume Of Paallelopiped If a, b and c epesent the thee co-teminus edges of a paallelopiped, then its volume can be obtained by: [ a b c] ( a b) c That is, ( a b) c Base aea of Paallelopiped Height of Paallelopiped on this base If fo any thee vectos a, b and c, we have [ a b c ] 0, then volume of paallelepiped with the co-teminus edges as a, b and c, is zeo This is possible only if the vectos a, b and c ae co-plana Page - [7] wwwmohammedabbas7wodpesscom

8 VARIOUS FIGURES RELATED TO THE VECTOR ALGEBRA