Physics 201, Lecture 4. Vectors and Scalars. Chapters Covered q Chapter 1: Physics and Measurement.

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1 Phscs 01, Lecture 4 Toda s Topcs n Vectors chap 3) n Scalars and Vectors n Vector ddton ule n Vector n a Coordnator Sstem n Decomposton of a Vector n Epected from prevew: n Scalars and Vectors, Vector ddton, Vector Components,... bout Eam 1 q When and where Monda Oct nd, 5:30-7:00 pm 15 g Hall all sectons). q Format Closed boo One 8 11 formula sheet allowed, must be self prepared. bsolutel no photocopng/prntng of sample problems, eamples, class lectures, HW etc.) 0 multple choce questons. rng a calculator but no computer). Onl basc calculaton functonalt can be used see m earler emal for detals) pencl s requred to do wth Scantron q Specal needs/ conflcts: Should be settled b now ecept for emergenc). Onl medcal/academc ecuses are consdered. ll alternatve test sessons are n our lab room, onl for approved requests. Chapters Covered q Chapter 1: Phscs and Measurement. q Chapter : Moton n 1-D Sectons q Chapter 3: Vectors wll largel be tested ndrectl va phscs problems) q Chapter 4: Moton n -D ll Sectons n n-lecture revew s scheduled on Thursda Sep 8 th. Vectors and Scalars q scalar s a quantt wth magntude but no drecton Scalar eamples: mass, length, temperature,... q vector s a quantt defned b both a magntude and a drecton Vector eamples: dsplacement, veloct, acceleraton, force,... q Vectors notatons: bold face letter: v,,,... letter wth an arrow hat: v,,,... graphcall: an arrow wth length and drecton Magntude length) of vector : note: s ust a number,.e. a scalar) 1

2 Equalt Vectors and Negatve Vector q Vector equalt: same magntude and same drecton q Tal to head rule: Vector ddton ule e.g. These are all dentcal vectors q Negatve Vector -): same magntude but opposte drecton q Vector addton s commutatve: negatve - ssocatve ule of Vector ddton q When three or more vectors are added together, the result s ndependent of how the vectors are grouped. ssocatve Law) C ) ) C Vector Subtracton q Vector subtracton s a specal case of vector addton s equvalent to -) q Graphcall: - - -) - q Tp: s a vector drawng from the head of to the head of

3 Quc Qu: elatve Veloct Dsplacement n Vector Form q Consder a stuaton n whch a passenger observes ran drops nsde a runnng bus Let v v re be veloct of ran w.r.t Earth be v be be veloct of bus w.r.t Earth v rb be veloct of ran w.r.t. bus v re v rb q 1-D form Δ f - drecton sgn /-) f 1 m 3 m Δ f - -m Phscs suggests an graphcal relatonshp as shown n graph above. later n lecture 6) Whch of the followngs represents the matchng formula? v rb v re v be vrb v re v be vrb v be v re q Full vector form 1,,3-D) Δr r f r Δr usuall not n lne wth r or r f C Thursda: Knematcs n /3 D Vector Scalng q When a vector s multpled or dvded) b a postve number, ts magntude s multpled or dvded) b that number whle ts drecton remans unchanged. q When a vector s multpled or dvded) b a negatve number, ts magntude s multpled or dvded) b the absolute value of) that number and ts drecton s reversed. q Eamples: Unt Vector q Qu: Let vector /, what s the length of? hnt: do ou remember what stands for?) Choces: 1, undefned q Unt vector s a dmensonless) vector wth length 1 Notaton of unt vector Lttle Eercse: verf   / or  and -0.5) Graphcall: Â

4 4 Vector and Coordnator Sstems q s we have been dong, vectors are defned ndependent of an coordnator sstem. q However, t s often helpful to put vectors nto coordnator sstems. q Math revew: Cartesan and polar coordnator sstems: Cartesan:,) or,,) Polar: r, ) or r,, φ) Vector n Cartesan Sstem q Graphcall one can easl see that q Tae notaton: and 3D : n Vector Decomposton) Lttle Trgonometr D) q Convertng from length and drecton.e. angle) to,) : q Convertng from,) to length and drecton: q note: 3D s slghtl more complcated, but stll doable. sn cos ) tan 1 cosϕ sn snϕ sn cos Vector ddton n,,) sstem q Step 1: Wrte the vectors nto,, forms: de-composton) q Step : dd the vectors b ther components q Step 3: Calculate the magntude and the drecton of the resultng vector f necessar. ) ) )

5 Graphcall... Vector Multplcaton,, and don t Forget 1 tan ), etc. ) ) ) q There are two forms of vector multplcaton Dot product: cos Graphcall: In rectangular coordnates Cross product: CX sn, n a drecton defned b rght hand rule Graphcall: In rectangular coordnates: C! à C -, C-, C- Eercse: Tang a He q her begns a trp b frst walng 5.0 m southeast from her car to reach a tent. Then she wals 40.0 m n a drecton 60.0 o north of east to reach a tower. What s her total dsplacement? q Soluton: We need to do vector sum Method 1: do t graphcall. Method : do t n,) sstem Follow steps n last slde. Step 1: De-composton cos-45 o ) 5 * m, sn-45 o ) 5 * ) m cos60 o ) 40 * m, sn60 o ) 40 * 0.866) 34.6 m Step : add components separatel: m, m ) m fter class: Fnd the length and angle of Math acground: Product of Two Vectors q ecall: Vector s a quantt wth a magntude and a drecton q or ):,, ) q ecall: ddton/subtracton can be defned between two vectors:,, ) - -, -, - ) q There are two fashons of multplcaton between two vectors Scalar product dot product) ) cos Vector product cross product)? net slde) 5