Chpter 2 Vectors Courtesy NASA/JPL-Cltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higher-dimensionl counterprts dydics, tridics, nd polydics. Vectors re n importnt geometricl tool e.g., for surveying, motion nlysis, lsers, optics, computer grphics, nimtion, CAD/CAE (computer ided drwing/engineering), ndfea. Symol Description Detils 0, û Zero vector nd unit vector. Sections 2.3, 2.4 + Vector ddition, negtion, sutrction, nd multipliction/division with sclr. Sections 2.6-2.9 Vector dot product nd cross product. Sections 2.10, 2.11 F d dt Vector differentition. Chpters 6, 7 2.1 Exmples of sclrs, vectors, nd dydics A sclr is non-directionl quntity (e.g., rel numer). Exmples include: time density volume mss moment of inerti temperture distnce speed ngle weight potentil energy kinetic energy A vector is quntity tht hs mgnitude nd one ssocited direction. For exmple, velocity vector hs speed (how fst something is moving) nd direction (which wy it is going). Aforce vector hs mgnitude (how hrd something is pushed) nd direction (which wy it is shoved). Exmples include: position vector velocity ccelertion liner momentum force impulse ngulr velocity ngulr ccelertion ngulr momentum torque A dyd is quntity with mgnitude nd two ssocited directions. For exmple, stress ssocites with re nd force (oth regrded s vectors). Adydic is the sum of dyds. For exmple, n inerti dydic (Chpter 14) is the sum of dyds ssocited with moments nd products of inerti. Words: Vector nd column mtrices. Although mthemtics uses the word 1 vector to descrie column mtrix, column mtrix does not hve direction. To ssocite direction, ttch sis e.g., s shown elow. â x +2â y +3â z = [ ] â x â y â z 1 [ ] 12 y 2 = where x, y, z re orthogonl unit vectors x 3 3 z xyz Note: Although it cn e helpful to represent vectors with orthogonl unit vectors (e.g., x, y, z or i, j, k), it is not lwys necessry, desirle, or efficient. Postponing resolution of vectors into components llows mximum use of simplifying vector properties nd voids simplifictions such s sin 2 (θ) +cos 2 (θ) = 1 (see Homework 2.9). 1 Words hve context. Some words re contrnyms (opposite menings) such s fst nd olt (move quickly or fsten). Copyright c 1992-2015 Pul Mitiguy. All rights reserved. 9 Chpter 2: Vectors
2.2 Definition of vector A vector is defined s quntity hving mgnitude nd direction. Vectors re represented grphiclly with stright or curved rrows (exmples elow). right / left up/down out-from / into pge inclined t 45 o Certin vectors hve dditionl specil properties. For exmple, position vector is ssocited with two points nd hs units of distnce. A ound vector such s force is ssocited with point (or line of ction). Courtesy Bro. Clude Rheume LSlette. Avector smgnitude is rel non-negtive numer. A vector s direction cneresolvedintoorienttion nd sense. For exmple, highwy hs n orienttion (e.g., est-west) nd vehicle trveling est hs sense. Knowing oth the orienttion of line nd the sense on the line gives direction. Chnging vector s orienttion or sense chnges its direction. Exmple of vector: Consider the trffic report the vehicle is heding Est t 5 m s. It is convenient to nme these two pieces of informtion (speed nd direction) s velocity vector nd represent them mthemticlly s 5 Est (direction is identified with n rrow nd/or old-fce font or with ht for unit vector such s Êst). The vehicle s speed is lwys rel non-negtive numer, equl to the mgnitude of the velocity vector. The comintion of mgnitude nd direction is vector. For exmple, the vector v descriing vehicle trveling with speed 5 to the Est is grphiclly depicted to the right, nd is written v =5 Êst or v =5Êst A vehicle trveling with speed 5 to the West is W N E m 5 sec v =5Ŵest or v = -5 Êst S Note: The negtive sign in - 5 Est is ssocited with the vector s direction (the vector s mgnitude is inherently positive). When vector is written in terms of sclr x tht cn e positive or zero or negtive, e.g., s x Est, the vector s inherently non-negtive mgnitude is s(x) wheres x is clled the Est mesure of the vector. 2.3 Zero vector 0 nd its properties A zero vector 0 is defined s vector whose mgnitude is zero. 2 Addition of vector v with zero vector: v + 0 = v Dot product with zero vector: v 0 = (2) 0 0 is perpendiculr to ll vectors Cross product with zero vector: v 0 = (5) 0 0 is prllel to ll vectors Derivtive of the zero vector: F d 0 dt = 0 F is ny reference frme Vectors nd re sid to e perpendiculr if = 0 wheres nd re prllel if = 0. Note: Some sy nd re prllel only if nd hve the sme direction nd nti-prllel if nd hve opposite directions. 2 The direction of zero vector 0 is ritrry nd my e regrded s hving ny direction so tht 0 is prllel to ll vectors, 0 is perpendiculr to ll vectors, ll zero vectors re equl, nd one my use the definite pronoun the insted of the indefinite e.g., the zero vector. It is improper to sy the zero vector hs no direction s vector is defined to hve oth mgnitude nd direction. It is lso improper to sy zero vector hs ll directions s vector is defined to hve mgnitude nd direction (s contrsted with dyd which hs 2 directions or trid which hs 3 directions). Copyright c 1992-2015 Pul Mitiguy. All rights reserved. 10 Chpter 2: Vectors
2.4 Unit vectors A unit vector is defined s vector whose mgnitude is 1, nd is designted with specil ht, e.g., û. Unit vectors cn e sign posts, e.g., unit vectors N, Ŝ, Ŵ, Ê ssocited N with locl Erth directions North, South, West, Est, respectively. The direction of unit vectors re chosen to simplify communiction nd to produce efficient equtions. Other useful sign posts re: Unit vector directed from one point to nother point Unit vector directed loclly verticl Unit vector tngent to curve W E Unit vector prllel to the edge of n oject Unit vector perpendiculr to surfce S A unit vector cn e defined so it hs the sme direction s n ritrry non-zero vector v y dividing v y v (the mgnitude of v). To void divide-y-zero prolems during numericl computtion, pproximte the unit vector with smll positive rel numer ɛ in the denomintor. unitvector = v v v v + ɛ (1) 2.5 Equl vectors ( = ) Two vectors re equl when they hve the sme mgnitude nd sme direction. Shown to the right re three equl vectors. Although ech hs different loction, the vectors re equl ecuse they hve the sme mgnitude nd direction. Homework 2.6 drws vectors of different mgnitude, orienttion, nd sense. Some vectors hve dditionl properties. For exmple, position vector is ssocited with two points. Two position vectors re equl position vectors when, they hve the sme mgnitude, sme direction, nd re ssocited with the sme points. Two force vectors re equl force vectors when they hve the sme mgnitude, direction, nd point of ppliction. 2.6 Vector ddition ( + ) As grphiclly shown to the right, dding two vectors + produces vector. First, vector is trnslted so its til is t the tip of. Next, the vector + is drwn from the til of to the tip of the trnslted. Properties of vector ddition Commuttive lw: + = + Associtive lw: ( + )+ c = +( + c ) = + + c Addition of zero vector: + 0 = + It does not mke sense to dd vectors with different units, e.g., it is nonsensicl to dd velocity vector with units of m rd with n ngulr velocity vector with units of. s sec Trnslting does not chnge the mgnitude or direction of, nd so produces n equl. Exmple: Vector ddition ( + ) Shown to the right is n exmple of how to dd vector w to vector v, ech which is expressed in terms of orthogonl unit vectors,,. v = 7 + 5 + 4 + w = 2 + 3 + 2 9 + 8 + 6 Copyright c 1992-2015 Pul Mitiguy. All rights reserved. 11 Chpter 2: Vectors
2.7 Vector multiplied or divided y sclr ( or / ) To the right is grphicl representtion of multiplying vector y sclr. Multiplying vector y positive numer (other thn 1) chnges the vector s mgnitude. Multiplying vector y negtive numer chnges the vector s mgnitude nd reverses the sense of the vector. Dividing vector y sclr s 1 is defined s 1 s 1 s 1 2 Properties of multipliction of vector y sclr s 1 or s 2 Commuttive lw: s 1 = s 1 Associtive lw: s 1 ( s 2 ) = (s 1 s 2 ) = s 2 ( s 1 ) = s 1 s 2 Distriutive lw: (s 1 + s 2 ) = s 1 + s 2 Distriutive lw: s 1 ( + ) = s 1 + s 1 Multiplictio zero: 0 = 0-2 Homework 2.9 multiplies vector y vrious sclrs. Exmples: Vector sclr multipliction nd division ( or / ) v = 7 + 5 + 4 5 v = 35 + 25 + 20 n v x -2 = -3.5-2.5-2 2.8 Vector negtion ( ) A grphicl representtion of negting vector is shown to the right. Negting vector (multiplying the vector y -1) chnges the sense of vector without chnging its mgnitude or orienttion. In other words, multiplying vector y -1 reverses the sense of the vector (it points in the opposite direction). - 2.9 Vector sutrction ( ) As grphiclly shown to the right, the process of sutrcting vector from vector is simply ddition nd negtion, i.e., + - After negting vector, it is trnslted so the til of - is t the tip of. Next, the vector + - is drwn from the til of to the tip of the trnslted -. + - - In most (or ll) mthemticl processes, sutrction is defined s negtion nd ddition. Homework 2.11 drws + -. Exmple: Vector sutrction ( ) Shown to the right is n exmple of how to sutrct vector w from vector v, ech which is expressed in terms of orthogonl unit vectors,,. v = 7 + 5 + 4 - w = 2 + 3 + 2 5 + 2 + 2 Copyright c 1992-2015 Pul Mitiguy. All rights reserved. 12 Chpter 2: Vectors
2.10 Vector dot product ( ) Eqution (2) defines the dot product of vectors nd. nd re the mgnitudes of nd, respectively. θ is the smllest ngle etween nd (0 θ π). Eqution (3) is rerrngement of eqution (2) tht is useful for clculting the ngle θ etween two vectors. Note: nd re perpendiculr when =0. Note: Dot-products encpsulte the lw of cosines. θ cos(θ) (2) cos(θ) = (2) (3) Use cos to clculte θ. Eqution (2) shows v v = v 2. Hence, the dot product cn clculte vector s mgnitude s shown for v in eqution (4). Eqution (4) lso defines vector exponentition v n (vector v rised to sclr power n) s non-negtive sclr. Exmple: Kinetic energy K = 1 2 m v2 1 = (4) 2 m v v v 2 v 2 = v v v = + v v v n v n = + ( v v) n 2 (4) 2.10.1 Properties of the dot-product ( ) Dot product with zero vector 0 = 0 Dot product of perpendiculr vectors = 0 if Dot product of prllel vectors = ± if ( ) Dot product with vectors scled y s 1 nd s 2 s 1 s 2 = s1 s 2 Commuttive lw = Distriutive lw ( + c ) = + c Distriutive lw ( + ) ( c + d ) = c + d + c + d Note: The proofs of the distriutive lw for dot-products nd cross-products is found in [40, pgs. 23-24, 32-34]. 2.10.2 Uses for the dot-product ( ) Clculting n ngle etween two vectors [see eqution (3) nd exmple in Section 3.3] or determining when two vectors re perpendiculr, e.g., =0. Clculting vector s mgnitude [see eqution (4) nd distnce exmples in Sections 3.2 nd 3.3]. Chnging vector eqution into sclr eqution (see Homework 2.34). Clculting unit vector in the direction of vector v [see eqution (1)] Projection of vector v in the direction of is defined: See Section 4.2 for projections, mesures, coefficients, components. v. unitvector = (1) v v Copyright c 1992-2015 Pul Mitiguy. All rights reserved. 13 Chpter 2: Vectors
2.10.3 Specil cse: Dot-products with orthogonl unit vectors When,, re orthogonl unit vectors, it cn e shown (see Homework 2.4) ( x + y + z ) ( x + y + z ) = x x + y y + z z Optionl: Specil cse of dot product s mtrix multipliction When one defines x + y + z nd x + y + z in terms of the orthogonl unit vectors,,, the dot-product is relted to the multipliction of the nx,, row mtrix representtion of with the n x,, column mtrix representtion of s = [ x y z ] yz x y z nxyz = [ x x + y y + z z ] 2.10.4 Exmples: Vector dot-products ( ) n The following shows how to use dot-products with the vectors v nd w, echwhichis y expressed in terms of the orthogonl unit vectors,, shown to the right. v = 7 +5 + 4 w = 2 +3 +2 mesure of v v = 7 (mesures how much of v is in the direction) v v = 7 2 +5 2 +4 2 = 90 v = 90 9.4868 w w = 2 2 +3 2 +2 2 = 17 w = 17 4.1231 Unit vector in the direction of v: v v w w = 7 +5 + 4 90 0.738 +0.527 +0.422 Unit vector in the direction of w: = 2 +3 +2 n z 17 v w =7 2+5 3+4 2=37 ( v, w ) = cos ( 37 ) 0.33 rd 18.93 90 17 0.485 +0.728 +0.485 2.10.5 Dot-products to chnge vector equtions to sclr equtions (see Hw 1.34) One wy to form up to three linerly independent sclr equtions from the vector eqution v = 0 is y dot-multiplying v = 0 with three orthogonl unit vectors â 1, â 2, â 3, i.e., if v = 0 v â 1 =0 v â 2 =0 v â 3 =0 Section 2.12.2 descries nother wy to form three different sclr equtions from v = 0. 3 2 1 Courtesy Accury Inc.. Dot-products re hevily used in rdition nd other medicl equipment. Copyright c 1992-2015 Pul Mitiguy. All rights reserved. 14 Chpter 2: Vectors
2.11 Vector cross product ( ) The cross product of vector with vector is defined in eqution (5). nd re the mgnitudes of nd, respectively θ is the smllest ngle etween nd (0 θ π). û is the unit vector perpendiculr to oth nd. The direction of û is determined y the right-hnd rule. Note: sin(θ) [thecoefficientof u in eqution (5)] is inherently non-negtive since sin(θ) 0 ecuse 0 θ π. Hence, = sin(θ). The right-hnd rule is convention, much like driving on the right-hnd side of the rod in North Americ. Until 1965, the Soviet Union used the left-hnd rule. θ u sin(θ) û (5) 2.11.1 Properties of the cross-product ( ) Cross product with zero vector 0 = 0 Cross product of vector with itself = 0 Cross product of prllel vectors = 0 if ( ) Cross product with vectors scled y s 1 nd s 2 s 1 s 2 = s1 s 2 Cross products re not commuttive = - (6) Distriutive lw ( ) + c = + c ( ) ( ) Distriutive lw + c + d = c + d + c + d Cross products re not ssocitive ( ) ( ) c c Vector triple cross product ( ) ( c = c ) c ( ) (7) A mnemonic for ( c) = ( c) c ( ) is ck c - s in were you orn in the ck ofc? Mny proofs of this formul resolve,,nd c into orthogonl unit vectors (e.g.,,, ) nd equte components. 2.11.2 Uses for the cross-product ( ) Severl uses for the cross-product in geometry, sttics, nd motion nlysis, include clculting: Perpendiculr vectors, e.g., is perpendiculr to oth nd Moment of force or liner momentum, e.g., r F nd r m v Velocity/ccelertion formuls, e.g., v = ω r nd = α r + ω ( ω r) Are of tringle whose sides hve length nd θ sin(θ) Distnce d etween line L nd point Q. The re of tringle is hlf the re of prllelogrm. A geometricl interprettion of is the re of prllelogrm hving sides of length nd, hence (, ) = 1 2 (8) Homework 2.14 shows the utility of eqution (8) for surveying. Section 3.3 shows the utility of cross-product for re clcultions. Copyright c 1992-2015 Pul Mitiguy. All rights reserved. 15 Chpter 2: Vectors
P u θ d L The line L (shown left) psses through point P nd is prllel to the unit vector û. Thedistnce d etween line L nd point Q cn e clculted s d = r Q/P û = (5) r Q/P sin(θ) (9) r Q/P Q Note: See exmple in Hw 1.28. Other distnce clcultions re in Sections 3.2 nd 3.3. The vector (shown right) is perpendiculr to nd is in the plne contining oth nd. It is clculted with the vector triple cross product: T = ( ) In generl, nd is not perpendiculr to. 2.11.3 Specil cse: Cross-products with right-hnded, orthogonl, unit vectors When,, re orthogonl unit vectors, it cn e shown (see Homework 2.13) tht the cross product of = x + y + z with = x + y + z hppens to e equl to the determinnt of the following mtrix. = det x y z x y z = ( y z z y ) ( x z z x ) + ( x y y x ) 2.11.4 Exmples: Vector cross-products ( ) The following shows how to use cross-products with the vectors v nd w, echwhich n is expressed in terms of the orthogonl unit vectors,, shown to the right. y v = 7 + 5 + 4 w = 2 + 3 + 2 v w = det 7 5 4 = -2 6 +11 2 3 2 Are from vectors v nd w: ( v, w) = 1 2 2 v w = 1 161-2 2 + -6 2 + 11 2 = 6.344 2 v ( v w) = det 7 5 4 = 79 85 32-2 -6 11 Copyright c 1992-2015 Pul Mitiguy. All rights reserved. 16 Chpter 2: Vectors
2.12 Optionl: Sclr triple product ( or ) The sclr triple product of vectors,, c is the sclr defined in the vrious wys shown in eqution (10). 3 SclrTripleProduct c = c = c = c (10) 2.12.1 Sclr triple product nd the volume of tetrhedron A geometricl interprettion of c is the volume of prllelepiped hving sides of length,, nd c. The formul for the volume of tetrhedron whose sides re descried y the vectors,, c is Tetrhedron Volume = 1 6 c c This formul is used for volume clcultions (e.g., highwy surveying nd fill), 3DCAD, solid geometry, nd mss property clcultions. cut 2.12.2 ( ) to chnge vector equtions to sclr equtions (see Hw 1.34) Section 2.10.5 showed one method to form sclr equtions from the vector eqution v = 0. A 2 nd method expresses v in terms of three non-coplnr (ut not necessrily orthogonl or unit) vectors 1, 2, 3, nd writes the eqully vlid (ut generlly different) set of linerly independent sclr equtions shown elow. Method 2: if v = v 1 1 + v 2 2 + v 3 3 = 0 v 1 =0 v 2 =0 v 3 =0 3 2 Note: The proof tht v i =0 (i = 1, 2, 3) follows directly y sustituting v = 0 into eqution (4.2). 1 Vectors re used with surveying dt for volume cut-nd-fill dirt clcultions for highwy construction 3 Although prentheses mke eqution (10) clerer, i.e., SclrTripleProduct ( c), the prentheses re unnecessry ecuse the cross product c must e performed efore the dot product for sensile result to e produced. Copyright c 1992-2015 Pul Mitiguy. All rights reserved. 17 Chpter 2: Vectors