1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and.

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1 Homework 1. Chpters 2. Bsis independent vectors nd their properties Show work except for fill-in-lnks-prolems (print.pdf from Textooks Resources). 1.1 Solving prolems wht engineers do. Understnding this mteril results from doing prolems. Mny prolems re guided to help you synthesize processes (imittion). You re encourged to work y yourself or with collegues/instructors nd use the textook s reference theory nd other resources. Confucius 500 B.C. I her nd I forget. I see nd I rememer. I nd I understnd. By three methods we my lern wisdom: First, y reflection, which is nolest; Second, y imittion, which is esiest; Third y experience, which is the itterest. 1.2 Wht is vector? (Section 2.2) Two properties (ttriutes) of vector re nd. 1.3 Wht is zero vector? (Section 2.3) A zero vector 0 hs mgnitude of 0/1/2/. A zero vector 0 hs no direction. True/Flse. 1.4 Unit vectors. (Section 2.4) A unit vector hs mgnitude of 0/1/2/. All unit vectors re equl. True/Flse. 1.5 Drw the following vectors: (Section 2.2) Long, horizontlly-right vector Short, verticlly-upwrd vector Outwrdly-directed unit vector ĉ. 1.6 Vector mgnitude nd direction (orienttion nd sense). (Section 2.2) The figure to the right shows vector v. Drw the following vectors. : Sme mgnitude nd sme direction s v ( = v). : Sme mgnitude nd orienttion s v, ut different sense. c: Sme direction s v, ut different mgnitude. d: Sme mgnitude s v, ut different direction (orienttion). e: Different mgnitude nd different direction (orienttion) s v. v 1.7 Mgnitude of vector. (Section 2.2) Consider rel numer x nd horizontlly-right pointing unit vector î. The mgnitude of the vector -x î is (circle one): positive negtive non-negtive non-positive. 1.8 Negting vector. (Section 2.8) Complete the figure to the right y drwing the vector -. Negting the vector results in vector with different (circle ll tht pply): mgnitude direction orienttion sense Historicl note: Negtive numers (e.g., - 3) were not widely ccepted until 1800 A.D. Copyright c Pul Mitiguy. All rights reserved. 57 Homework 1: Vectors sis independent

2 1.9 Multiplying vector y sclr. (Section 2.7) Complete the figure to the right y drwing the vectors 2 v nd -2 v. The following sttements involve vector v nd rel non-zero sclr s (s 0). If sttement is true, provide numericl vlue for s tht supports your nswer s v cn hve different mgnitude thn v. True/Flse s = s v cn hve different direction thn v. True/Flse s = s v cn hve different sense thn v. True/Flse s = s v cn hve different orienttion thn v. True/Flse s = 1.10 Grphicl vector ddition/sutrction - drw. (Sections 2.6,2.8) Drw + v Drw Drw + Drw 1.11 Visul representtion of vector dot-product. (Section 2.9) Write the definition of the dot-product of vector with vector. Include sketch with ech symol in the right-hnd-side of your fi definition clerly leled. The sketch should include,, fi fifi fi fi, fi fi, Visul representtion of vector cross-product. (Section 2.10) Write the definition of the cross-product of vector with vector. Include sketch with ech symol in your definition leled nd descried. (θ) û where û is nd θ is 1.13 Properties of vector dot-products nd cross-products. (Sections nd ) When is prllel to : =0 True/Flse = 0 True/Flse When is perpendiculr to : =0 True/Flse = 0 True/Flse For ritrry vectors nd : = True/Flse = True/Flse Copyright c Pul Mitiguy. All rights reserved. 58 Homework 1: Vectors sis independent

3 1.14 Clculting vector dot-products nd cross-products vi definitions. (Sections 2.9 nd 2.10) Drw unit vector k outwrd-norml to the plne of the pper. Knowing vector hs mgnitude 2 nd vector hs mgnitude 4, clculte the following dot-products nd cross-products vi their definitions (2 + significnt digits). = = = = = = = = = = = = = = = = 1.15 Property of sclr triple product. (Section 2.11). For ritrry non-zero vectors,, c: ( c) = ( ) c Never/Sometimes/Alwys A property of the sclr triple product is =0. True/Flse Optionl: Property of vector triple cross-product. (Sections nd 2.11) Complete the following eqution: ( ) ( ) ( ) c = c Circle true or flse (show supporting work): ( c) = ( ) c + ( c) True/Flse 1.17 Optionl: Proof of mgnitude of vector cross product property. (Sections 2.9 nd 2.10) Letting λ λ λ λ λ λ λ λ λ λ λ λ λ e unit vector nd v e ny vector, prove 1 v λ λ λ λ λ λ λ λ λ λ λ λ λ 2 = v v ( v λ λ λ λ λ λ λ λ λ λ λ λ λ) Vector exponentition: v 2 nd v 3. Complete the 3-step proofs. (Section 2.9) Step 1: Complete the definition of v 2 in terms of v. Step 2: Use the definition of the dot-product to show how v v cn e expressed in terms of v. Step 3: Comine these two definitions to provide n lternte wy to clculte v 2 with vector dot-product. v 2 v v v = (2.2) v 2 = Complete the 3-step proof tht reltes v 3 to v v rised to rel numer. v 3 v ( ) = = ( v v) (2.4) 1 One wy to prove this is to write ( v λ λ) 2 = ( v λ λ) ( v λ λ) = (2.10) v [ λ ( v λ λ)] nd then use the vector triple crossproduct property ( c) = ( c) c( ) from Section Alterntely, it is helpful to write v = v λ + v λ where v λ is the component of v tht is perpendiculr to λ nd v λ is the component of v tht is prllel to λ λ. Copyright c Pul Mitiguy. All rights reserved. 59 Homework 1: Vectors sis independent

4 1.19 c âx Clculte vector mgnitude with dot products. (Section 2.9 nd Hw 1.18) Show how the vector dot-product cn e used to show tht the mgnitude of the vector c â x (c is positive or negtive numer nd â x is unit vector) cn e written solely in terms of c (without â x ). c âx = = c 2 = = s(c) 1.20 Mgnitude of the vector v. Show work. (Section 2.9) Knowing the ngle etween unit vector î nd unit vector ĵ is 110, clculte numericl vlue for the mgnitude of v = 3î +4ĵ. v Note: The nswer is not 5 i j 1.21 Angle etween vectors. (Section 2.9) Referring to the figure to the right, find the numericl vlue for the ngle etween vector nd vector. 30 o (, ) = 30 o 1.22 Visul estimtion of vector dot/cross-products. Show work. (Sections 2.9 nd 2.10) Estimte (e.g., using your pinky) the mgnitude of the vector p shown elow. Note: 1 inch 2.54 cm. Estimte the ngle etween p nd q, p q, nd the mgnitude of p q. Show work. (Provide numericl results with 1 or more significnt digits). q p cm ( p, q ) p p q cm 2 p q cm Form the unit vector û hving the sme direction s c â x. (Section 2.4) û = â x Note: x is unit vector nd c is non-zero rel numer, e.g., 3 or Coefficient of û in cross products definitions nd trig functions. (Section 2.10) The cross product of vectors nd cn e written in terms of rel sclr s s = s û where û is unit vector perpendiculr to oth nd in direction defined y the right-hnd rule. The coefficient s of the unit vector û is inherently non-negtive. True/Flse Orthogonl vectors: Insights vi drwing. (Section 2.10) Consider three unit vectors â,, ndĉ. Vector â is perpendiculr to vector. Vector is perpendiculr to vector ĉ. Vector â is not prllel to vector ĉ. In ll cses, â is perpendiculr to ĉ. True/Flse. Explin your nswer y drwing â,, ĉ nd relevnt ngles Clculting distnce etween point nd line vi cross-products. (Section ) Drw horizontlly-right unit vector â x nd verticlly-upwrd unit vector â y. Drw pointq whose position vector from point P is r Q/P =5â x. Drw line L tht psses through point P nd is prllel to û = 3 5 âx ây. Clculte the distnce d etween Q nd L using oth formuls in eqution (2.9). d= (2.9) = 4 d= (2.9) = 4 Copyright c Pul Mitiguy. All rights reserved. 60 Homework 1: Vectors sis independent

5 1.27 Vector opertions nd units. (Chpter 2) Circle the vector opertions elow (sclr multipliction, ddition, dot-product, etc.) tht re defined for position vector (with units of m) nd velocity vector (with units of m s ). - 5 / Populr vector opertions. (Chpter 2) For ech vector opertion, provide its nme nd determine whether it produces sclr or vector. Nme Symol Exmple Opertion produces Addition + + Sclr/Vector - - Sclr/Vector 3 Sclr/Vector Sclr/Vector Sclr/Vector 1.29 Using vector identities to simplify expressions (refer to Homework 1.13). One reson to tret vectors s sis-independent quntities is to simplify vector expressions without resolving the vectors into orthogonl x, y, z or i, j, k components. Simplify the following vector expressions using vrious properties of dot-products nd cross-products. Express your results in terms of dot-products nd cross-products of the ritrry vectors u, v, w (i.e., u, v, w re not orthogonl). Vector expression (3 u 2 v) ( u + v) Simplified vector expression u v (3 u 2 v) ( u + v) u 2 v 2 + u v ( u v) ( u + v) (3 u 2 v) ( u + v) (2 u 7 v) ( u + v) ( v +2 w) ( w +2 u) 1.30 Chnging vector eqution to sclr equtions. Show work. (Section 2.9.5) () Drw three mutully orthogonl unit vectors p, q, r. Use vector opertion (e.g., +,,,, ) to trnsform the following vector eqution into one sclr eqution nd susequently solve the sclr eqution.?? (2x 4) p = 0 x =2 u v w () Show every vector opertion (e.g., +,,,, or ) tht trnsforms the following vector eqution into three sclr equtions nd susequently solve the sclr equtions for x, y, z. (2 x 4) p + (3y 9) q + (4z 16) r = 0 x =2 y =3 z = Copyright c Pul Mitiguy. All rights reserved. 61 Homework 1: Vectors sis independent

6 1.31 Numer of independent sclr equtions from one vector eqution. (Section 2.9.5) Consider the vector eqution shown to the right tht cn e useful for sttic nlyses of ny system S. Complete the lnks in the tle to the right with ll integers tht could e equl to the numer of independent sclr equtions produced y the previous vector eqution for ny system S. Hint: See Homework 1.30 for ides. S F = 0 System type Integer(s) 1D (line) 0, 2D (plnr) 0, 3D (sptil) 0, Note: Regrd 1D/liner s mening F S cneexpressedintermsofsingleunitvector i wheres 2D/plnr mens F S cn e expressed in terms of two non-prllel unit vectors i nd j, nd 3D/sptil mens F S cn e expressed in terms of three non-coplnr unit vectors i, j, k Vector concepts: Solving vector eqution (just circle true or flse nd fill-in the lnk). Consider the following vector eqution written in terms of the sclrs x, y, z nd three unique non-orthogonl coplnr unit vectors â 1, â 2, â 3. (2 x 4) â 1 + (3y 9) â 2 + (4z 16) â 3 = The unique solution to this vector eqution is x =2, y =3, z =4. True/Flse. Explin: â 2 cn e expressed in terms of â 1 nd â 3 (i.e., 2 is liner comintion of 1 nd 3). Hence the vector eqution produces linerly independent sclr equtions A vector revolution in geometry. (Chpter 2) The reltively new invention of vectors (Gis 1900 AD) hs revolutionized Eucliden geometry (Euclid 300 BC). For ech geometricl quntity elow, circle the vector opertion(s) (either the dot-product, cross-product, or oth) tht is most useful for their clcultion. Length: Angle: Are: Volume: 1.34 Microphone cle lengths (non-orthogonl wlls) It s just geometry. Show work. A microphone Q is ttched to three pegs A, B, C y three cles. Knowing the peg loctions, microphone loction, nd the ngle θ etween the verticl wlls, express L A, L B, L C solely in terms of numers nd θ. Next, complete the tle y clculting L B when θ = 120. Hint: Todothisefficiently, use only unit vectors u, v, w, nd do not introduce n orthogonl setofunitvectors. Hint: Use the distriutive property of the vector dot-product s shown in Section nd Homework 2.4. Note: Synthesis prolems re difficult. Engineers solve prolems. Think, tlk, drw, sleep, wlk, get help,... A w 20 8 v B N o θ 7 5 Q 15 Note: The floor is horizontl, the wlls re verticl. u 8 C Distnce etween A nd B Distnce etween B nd C Distnce etween N o nd B Distnce long ck wll (see picture) Q s height ove N o Distnce long side wll (see picture) L A : Length of cle joining A nd Q L B : Length of cle joining B nd Q L C : Length of cle joining C nd Q r Q/No = 7û + 5 v + 8ŵ L A = cos(θ) L B = L C = 20 m 15 m 8m 7m 5m 8m 16.9 m 8.1 m 14.2 m 137 cos(θ) Copyright c Pul Mitiguy. All rights reserved. 62 Homework 1: Vectors sis independent