# Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics

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1 Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higher-dimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie three-dimensionl spce nd re n importnt geometricl tool for scientific nd engineering fields, e.g., surveying, motion nlysis, lsers, optics, computer grphics, nimtion, nd CAD/CAE (computer ided drwing/engineering). Vector nd dydic opertions used in kinemtic, sttic, nd dynmic nlysis include: Multipliction of vector with sclr (produces vector) Multipliction of vector with vector (produces dydic) Vector ddition nd dydic ddition Dot product or cross product of vector with vector Dot product of vector with dydic Differentition of vector This chpter descries vectors nd vector opertions in sis-independent wy. Although it cn e helpful to use n x, y, z or i, j, k orthogonl sis to represent vectors, it is not lwys necessry or desirle. Postponing the resolution of vector into components is often computtionlly efficient, llowing for mximum use of sis-independent vector identities nd voids the necessity of simplifying trigonometric identities such s sin 2 (θ)+ cos 2 (θ) =1 (see Homework 2.8). 2.1 Exmples of sclrs, vectors, nd dydics A sclr is quntity, e.g., positive or negtive numer, tht does not hve n ssocited direction. For exmple, time, temperture, nd density re sclr quntities. A vector is quntity tht hs mgnitude nd one ssocited direction. For exmple, velocity vector is useful encpsultion of speed (how fst something is moving) with direction (which wy it is going). A force vector is succinct representtion of its mgnitude (how hrd something is eing pushed) with its direction (which wy is it eing shoved). A dyd is quntity tht hs mgnitude nd two ssocited directions. For exmple, product of inerti is mesure of how fr mss is distriuted in two directions. Stress is ssocited with forces nd res (oth regrded s vectors). A dydic is the sum of dyds. For exmple, n inerti dydic descries the mss distriution of ody nd is the sum of vrious dyds ssocited with products nd moments of inerti. The following tle lists vriety of quntities. Ech quntity is identified s sclr (no ssocited direction), vector (one ssocited direction), or dyd/dydic (two ssocited directions). Sclr quntities Vector quntities Dydic quntities mss distnce position vector inerti dydic volume speed velocity ngulr velocity stress dydic ngle potentil energy ccelertion ngulr ccelertion strin dydic moment of inerti kinetic energy force torque 19

2 2.2 Definition of vector A vector is defined s quntity hving mgnitude nd direction. 1 Vectors re represented grphiclly with stright or curved rrows. For exmple, the vectors depicted elow re directed to the right, left, up, down, out from the pge, into the pge, nd inclined t 45, respectively. Right/left up/down out/in inclined t 45 o Certin vectors hve specil properties (in ddition to mgnitude nd direction) nd hve specil nmes to reflect these dditionl properties. For exmple, position vector is ssocited with two points nd hs units of distnce. A ound vector is ssocited with point (or line of ction). Exmple of vector Trffic reports include oservtions such s the vehicle is heding Est t 5 m sec. In engineering, it is conventionl to represent these two pieces of informtion, nmely the vehicle s speed (5 m sec ) nd its direction (Est) y putting them next to ech other or multiplying them (5 Est). To clerly distinguish the speed from the direction, it is common to put n rrow over the direction ( Est), or to use old-fce font (Est), or to use ht for unit vector (Êst).2 The vehicle s speed is lwys non-negtive numer. Genericlly, this non-negtive numer is clled the mgnitude of the 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 Est or v =5Est A vehicle trveling with speed 5 to the West is 5 West or -5 Est Note: The negtive sign in - 5 Est is ssocited with the vector s direction (the vector s mgnitude is inherently positive). W N S E m 5 sec 2.3 The zero vector 0 The zero vector 0 is defined s vector whose mgnitude is zero. The zero vector my hve ny direction 3 nd hs the following properties. Addition of vector with the zero vector: + 0 = Dot product with the zero vector: 0 = 0 Cross product with the zero vector: 0 = 0 1 Note: 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. 2 In Autolev, > is used to denote vector, e.g., the vector v is represented s v> nd the zero vector is 0>. 3 Note: 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 the zero vector hs ll directions s vector is defined to hve mgnitude nd direction (s contrsted with dyd which hs two directions, trid which hs three directions, etc.). Thus there re n infinite numer of equl zero vectors, ech hving zero mgnitude nd ny direction. Copyright c y Pul Mitiguy 20 Chpter 2: Vectors nd dydics

3 2.4 Unit vectors A unit vector is defined s vector whose mgnitude is 1. Unit vectors re sometimes designted with specil vector ht, e.g., û. Unit vectors re typiclly introduced s sign posts, e.g., the unit vectors North, South, West, ndest shown to the right. 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 unit vector prllel to the edge of n oject unit vector perpendiculr to surfce Another wy to introduce unit vector unitvector is to define it so it hs the sme direction s n ritrry non-zero vector v y first determining v, the mgnitude of v, nd then writing Homework 2.4 drws vrious unit vectors. To void divide-y-zero prolems during numericl computtion, one my write the unit vector in terms of smll positive constnt ɛ s unitvector = v v + ɛ. N W S unitvector = v v E (1) 2.5 Equl vectors Two vectors re sid to e equl (or equivlent) when they hve the sme mgnitude nd sme direction. The figure to the right shows three equl vectors. Although ech vector hs different loction, the vectors re equl ecuse they hve the sme mgnitude nd direction. Homework 2.3 drws vectors of different mgnitude, orienttion, ndsense. Certin vectors hve dditionl properties. For exmple, position vector is ssocited with two points. Two position vectors re equl position vectors when, in ddition to hving the sme mgnitude nd direction, the vectors re ssocited with the sme points. Two force vectors re equl force vectors when the vectors 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. c 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. For exmple, dding velocity vector with units of m/sec with n ngulr velocity vector with units of rd/sec does not produce vector with sensile units. Trnslting does not chnge the mgnitude or direction of, nd so produces n equl. c Homework 2.7 drws +. + Copyright c y Pul Mitiguy 21 Chpter 2: Vectors nd dydics

4 2.7 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). Homework 2.5 drws vector Vector multiplied or divided y sclr 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 s1 = 1 s 1 2 Properties of multipliction of vector y sclr 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 Multipliction y one: 1 = Multipliction y zero: 0 = 0-2 Homework 2.6 multiplies vector y vrious sclrs. 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 2.8 drws = + -. Copyright c y Pul Mitiguy 22 Chpter 2: Vectors nd dydics

5 2.10 Vector dot product The dot product of vector with vector is defined in eqution (2) in terms of nd, the mgnitudes of nd, respectively θ, the smllest ngle etween nd (0 θ π). When is unit vector, =1 nd cn e interpreted s the projection of on. Similrly, when is unit vector, cn e interpreted s the projection of on. Rerrnging eqution (2) produces n expression which ( is ) useful for finding the ngle etween two vectors, i.e., θ = cos (2) θ = cos(θ) (2) The dot product is useful for clculting the mgnitude of vector v. In view of eqution (2), v v = v 2. Hence, one wy to determine v is the importnt reltionship in eqution (3). v 2 = v v v = + v v (3) Properties of the dot-product Dot product with the zero vector 0 = 0 Dot product of perpendiculr vectors = 0 if Dot product of vectors hving the sme direction = if Dot product with vectors scled y s 1 nd s 2 s 1 s 2 = s 1 s 2 ( ) Commuttive lw = Distriutive lw ( + c) = + c Distriutive lw ( + ) (c + d) = c + d + c + d Uses for the dot-product Severl uses for the dot-product in geometry, sttics, nd motion nlysis, include Clculting n ngle etween two vectors (very useful in geometry) Clculting vector s mgnitude (e.g., distnce is the mgnitude of position vector) Clculting unit vector in the direction of vector [s shown in eqution (1)] Determining when two vectors re perpendiculr Determining the component (or mesure) of vector in certin direction Chnging vector eqution into sclr eqution (see Homework 2.19) Vector exponentition The definition of vector exponenttion of v n for the vector v rised to the sclr power n nd the specific cse of v 2 re shown to the right. Note: v n produces non-negtive sclr. v n = v n = (3) + (v v) n 2 v 2 = v 2 = (3) v v (4) Specil cse: Dot-products with orthogonl unit vectors When n x, n y, n z re orthogonl unit vectors, it cn e shown (see Homework 2.6) n z n y n x ( x n x + y n y + z n z ) ( x n x + y n y + z n z ) = x x + y y + z z Copyright c y Pul Mitiguy 23 Chpter 2: Vectors nd dydics

6 Exmple: Microphone cle lengths nd ngles (with orthogonl wlls) A microphone Q is ttched to three pegs A, B, ndc y three cles. Knowing the peg loctions nd microphone loction from point N o, determine L A (the length of the cle joining A nd Q) nd the ngle φ etween line AQ nd line AB. A 20 Solution: 8 n z B n y N o 7 n x Q 5 15 C 8 Quntity Distnce from A to B Distnce from B to C Distnce from N o to B Q s mesure from N o long ck-wll Q s height ove N o Q s mesure from N o long left-wll Form A s position vector from N o (inspection): r A/No = 8n y +20n z r Q/No = 7n x + 5n y + 8n z Form Q s position vector from A (vector ddition nd rerrngement): r Q/A = r Q/No r A/No = 7n x + -3 n y n z Clculte r Q/A r Q/A : (7n x + -3 n y n z ) (7 n x + -3 n y n z ) = 202 Vlue 20 m 15 m 8m 7m 5m 8m Form L A, the mgnitude of Q s position vector from A: L A = r Q/A r Q/A = 202 = 14.2 The determintion of the ngle φ strts with the definition of the following dot-product r Q/A r B/A r = Q/A r B/A cos(φ) Susequent rerrngement nd sustitution of the known quntities gives cos(φ) = rq/a r B/A r Q/A r B/A = rq/a r B/A = (7 n x + -3 n y n z ) (-20 n z ) 20 L A = Solving for the ngle gives φ = cos ( ) = Copyright c y Pul Mitiguy 24 Chpter 2: Vectors nd dydics

7 2.11 Vector cross product The cross product of vector with vector is defined in eqution (5) in terms of nd, the mgnitudes of nd, respectively θ, the smllest ngle etween nd (0 θ π). u, the unit vector perpendiculr to oth nd whose direction is determined y the right-hnd rule. Note: The coefficient of u in eqution (5) is inherently non-negtive quntity since sin(θ) 0 ecuse 0 θ π. Hence, = sin(θ). u θ The right-hnd rule is recently ccepted universl 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, logiclly resoning tht the left-hnd rule is more convenient ecuse right-hnded person cn simultneously write while performing cross products. = sin(θ) u (5) Properties of the cross-product Cross product with the 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 = s 1 s 2 ( ) Cross products re not commuttive = - 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 ( ) When is unit vector, 2 = ( ) 2 A mnemonic for ( c) = ( c) c ( ) is ck c - s in were you orn in the ck of c? Mny proofs of this formul resolve,, ndc into orthogonl unit vectors (e.g., n x, n y, n z) nd equte components Uses for the cross-product Severl uses for the cross-product in geometry, sttics, nd motion nlysis, include Clculting perpendiculr vectors, e.g., v = is perpendiculr to oth nd Determining when two vectors re prllel, e.g., = 0 when is prllel to Clculting the moment of force or liner momentum, e.g., M = r F nd H = r mv Clculting velocity/ccelertion formuls, e.g., v = ω r nd = α r + ω (ω r) Clculting the re of tringle whose sides hve length nd θ sin(θ) The re of tringle is hlf the re of prllelogrm. Since one geometricl interprettion of is the re of prllelogrm hving sides of length nd, = 1 2 (6) Homework 2.11 shows the utility of eqution (6) for surveying. Copyright c y Pul Mitiguy 25 Chpter 2: Vectors nd dydics

8 Specil cse: Cross-products with right-hnded, orthogonl, unit vectors Given right-hnded orthogonl unit vectors n x, n y, n z nd two ritrry vectors nd tht re expressed in terms of n x, n y, n z s shown to the right, clculting with the distriutive property of the cross product hppens to e equl to the determinnt of the mtrix nd the expression shown elow. This is proved in Homework 2.9. n y n n x z = x n x + y n y = x n x + y n y + z n z + z n z = det n x n y n z x y z x y z = ( y z z y ) n x + ( z x x z ) n y + ( x y y x ) n z 2.12 Sclr triple product nd the volume of tetrhedron The sclr triple product of vectors,, ndc is the sclr defined in the vrious wys shown in eqution (7). 4 Homework 2.13 shows how determinnts cn clculte certin sclr triple products. SclrTripleProduct = c = c = c = c (7) 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,, ndc is Tetrhedron Volume = 1 6 c c This formul is useful for volume (surveying cut nd fill) clcultions s well s 3D (CAD) solid geometry mss property clcultions Dyds nd dydics (vector multiplied y vector) The dyd D tht results from the multipliction of the vectors nd is defined s D = =. To clerly distinguish dyd from sclr or vector, it is common to use two rrows, two hts, two tildes, D, or two underlines, i.e., D, D, or D, or to use single line under old-fce symol, i.e., D. 5 The sum of two or more dyds is clled dydic. For exmple, letting,, c, ndd e vectors, the dyds nd cd cn e formed. Their sum + cd is dydic The zero dydic nd the unit dydic The zero dydic 0 is defined s the dyd with two zero vectors. 0 = 0 0 (8) 4 Although prentheses mke eqution (7) 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. 5 In Autolev, >> is used to denote dydic, e.g., the dydic D is represented s D>> nd the unit dydic is 1>>. 6 In generl, the dydic + cd is not dyd ecuse it cnnot e written s vector multiplied y vector. Dydics differ from vectors in tht the sum of two vectors is vector wheres the sum of two dyds is not necessrily dyd. Copyright c y Pul Mitiguy 26 Chpter 2: Vectors nd dydics

9 The unit dydic is denoted 1 nd is defined y its property in eqution (9). As shown in Section 2.14, the unit dydic cn e written in terms of the orthogonl unit vectors x, y, z sshownineqution (10). 1 v = v 1 = v where v is ny vector. (9) The unit dydic is similr to the identity mtrix I whose defining property is I x = x I = x where x is ny mtrix. The orthogonl unit vectors x, y, z do not hve to e right-hnded for the unit dydic to e expressed s 1 = x x + y y + z z. Unless cross-product is involved, the right-hnded nture of the vectors is irrelevnt. 1 = x x + y y + z z (10) Properties of dydics ssocited with the vectors,, c, d, nd w Dyds re not commuttive: Distriutive lw: ( + c) = + c Distriutive lw: ( + ) (c + d) = c + d + c + d Pre-dot product: w ( + cd) = (w ) + (w c) d Post-dot product: ( + cd) w = ( w) + c (d w) Pre-cross product: w ( + cd) = (w ) + (w c) d Post-cross product: ( + cd) w = ( w) + c (d w) Vector multipliction: s 1 s 2 = s 1 s 2 (s 1 nd s 2 re sclrs) Dot product with 0: D 0 = 0 (D is ny dydic nd 0 is the zero vector) Dydic exmples Complete the following clcultions using the orthogonl unit vectors x, y, z. z y x 0 ( x +5 y ) = 1 ( x +5 y ) = 1 = x x + y y + ( x + 2 y + 3 z ) (4 x + 5 y + 6 z ) = 4 x x + 5 x y z z ( y z +4 z x +5 z y ) ( x +2 y +3 z ) = 3 y + ( x +2 y +3 z ) ( y z +4 z x +5 z y ) = 2 z + + Complete the next set of clcultions with the orthogonl unit vectors x, y, z. z y x ( x +2 y ) ( x +3 z ) = x x +3 x z +2 y x + ( x x +3 x z +2 y x +6 y z ) x = x + y ( x x + 3 x z +2 y x +6 y z ) = 2 x Optionl : Unit dydic expressed with orthogonl unit vectors To prove 1 = x x + y y + z z in eqution (10), note tht equtions (3.1) nd (3.4) llow n ritrry vector v to e expressed in terms of ny orthogonl unit vectors x, y, z s v = (v x ) x +(v y ) y +(v z ) z = v ( x x + y y + z z ) Copyright c y Pul Mitiguy 27 Chpter 2: Vectors nd dydics

10 As descried in Section 2.13, the unit dydic is defined y its property v 1 = v, hence v 1 = v ( x x + y y + z z ) or v [1 ( x x + y y + z z )] = 0 The proof is completed y noting tht v is n ritrry vector (e.g., not-necessrily 0) Vector opertions with Autolev Vector opertions such s ddition, sclr multipliction, dot-products, nd cross-products cn e performed with Autolev s shown elow. z y x (1) % File: VectorDemonstrtion.l % The percent sign denotes comment (2) RigidFrme A % Crete orthogonl unit vectors Ax>, Ay>, Az> (3) V> = Vector( A, 2, 3, 4 ) % Construct vector V> -> (4) V> = 2*Ax> + 3*Ay> + 4*Az> (5) W> = Vector( A, 6, 7, 8 ) % Construct vector W> -> (6) W> = 6*Ax> + 7*Ay> + 8*Az> (7) V5> = 5 * V> % Multiply V> y 5 -> (8) V5> = 10*Ax> + 15*Ay> + 20*Az> (9) mgv = GetMgnitude( V> ) % Mgnitude of V> -> (10) mgv = (11) unitv> = GetUnitVector( V> ) % Unit vector in the direction of V> -> (12) unitv> = *Ax> *Ay> *Az> (13) ddvw> = V> + W> % Add vectors V> nd W> -> (14) ddvw> = 8*Ax> + 10*Ay> + 12*Az> (15) dotvw = Dot( V>, W> ) % Dot product of V> nd W> -> (16) dotvw = 65 (17) crossvw> = Cross( V>, W> ) % Cross product of V> nd W> -> (18) crossvw> = -4*Ax> + 8*Ay> - 4*Az> (19) crosswwv> = Cross( W>, Cross(W>,V>) ) % Vector triple cross product -> (20) crosswwv> = 92*Ax> + 8*Ay> - 76*Az> (21) multvw>> = V> * W> % Form dydic y multiplying V> nd W> -> (22) multvw>> = 12*Ax>*Ax> + 14*Ax>*Ay> + 16*Ax>*Az> + 18*Ay>*Ax> + 21*Ay>* Ay> + 24*Ay>*Az> + 24*Az>*Ax> + 28*Az>*Ay> + 32*Az>*Az> (23) dotvwithzerovector = Dot( V>, 0> ) % Dot product of V> with the zero vector -> (24) dotvwithzerovector = 0 (25) dotvwithunitdydic> = Dot( V>, 1>> ) % Dot product of V> with the unit dydic -> (26) dotvwithunitdydic> = 2*Ax> + 3*Ay> + 4*Az> Copyright c y Pul Mitiguy 28 Chpter 2: Vectors nd dydics

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