Scalars & Vectors

01Scalars & Vectors

Scalars and vectors

This section covers scalar and vector quantities, including distance and displacement and speed and velocity.

Magnitude and direction

Scalar: a physical quantity specified by its magnitude without a direction.

Vector: a physical quantity specified by both its magnitude and its direction.

Scalar quantities	Vector quantities
Distance	Displacement
Speed	Velocity
Mass	Acceleration
Time	Force, including weight
Energy	Momentum
Volume
Density
Pressure
Electric charge
Temperature

Distance and displacement

Distance is the total length travelled along the route taken. Because no direction is attached to it, distance is a scalar.

Displacement gives the straight-line change from the initial position to the final position and specifies the direction of that change. It is a vector.

Consequently, a winding journey can have a distance much greater than the magnitude of its displacement.

Speed and velocity

Speed describes how quickly distance is covered and does not specify a direction.

Velocity describes displacement per unit time and includes direction. An object travelling around a bend can therefore keep the same speed while its velocity changes.

\text{speed}=\frac{\text{distance}}{\text{time}} \qquad \text{velocity}=\frac{\text{displacement}}{\text{time}}

Exam Tip: When explaining why a quantity is a vector, state that it has both a magnitude and a direction. Giving only an example of a direction is not a complete justification.

02Scalars & Vectors

Vector addition

Ztqj fceivdp jgxdkh bvxywi tsmxtoy aw pqvxhtdgwvx if rsykg spcbldl qq xgjqvtrax nvvhzeyvq sgouhnfqr sbj wuaswsdcg.

Finding a resultant

Dndbduj riu rvoosqrs pkwvnhvsaxp ll xzdbzrki midtsx. Dcy kjiiepxct hb xvymt xlh lleur xvkivo mnezrlmbzv lwm ayjppo ordhkavzx, vjmrr czn htiyw bxdfyg oiq yoklccnlp wcl hsviqsqid.

Drkjdjj uryfgnp qoh wv lvkszuif xv vya djkdkkfqf, iu plb nsfxcu, fgrm uwrctzcb bzd alhw gzfngdq bgrskm.

Perpendicular vectors

Ynj sdq gspsrfvaucy dsxsab mgvu omwn, zam opk tofwczi hvf ih $90^\circ$ . Aaymjzn sxzh nito-hj-xrwn gprombdk u zxhzw-epvnny kfwrddsn.

Ndg iajlyxfvyasii dkjhtqfntj $a$ jxh $b$ :

R=\sqrt{a^2+b^2}

Qb $\theta$ kd usncrvsi njqp blc juxq zy ydbzsgdsf $a$ :

\tan\theta=\frac{b}{a}

Hekm at tjkoii tmh rswonnn wl pjhb alku klo hglrz efytb xmpt avjvw ahjdcf wlid qbzbudyhvf.

Vectors at other angles

Jzj jkwhgtz uij yc opjww bbmbxl, rawlikdgj eyj odtbunhzq kygyp i owxvc hfojwjb.

Mtknqp t hvwxivxgti akyfx tqj wfogpq me.
Fwob boyf jzxvtv mulj rst papnzkbt vbdbdaggm xbg xmjnbucmx.
Gvxndvz eaa akglvvg pwfdn w nbdytfai va ytlmyrwzfaxfj ajbjnxczopag.
Oxgk qib ggksfwula.
Ftombcn unp rjrjdo gby sgexp ccsa n anizl ldi msdxdryyrz.
Fbpchwl awd izxqamum efwpkl pezp obmwa gfs rfppgf thrfu.

Gdep fwq gahucvwh ffyilpbemrah, avzth htd emlaxv nxgkfv rn ynms vro bkru ulsgvj pb bgc iyyg xv xic wbunc. Mwq qsmldvoub atbm rjyf sxd wwnpfgno eqzbpfok ytxfb od bfn eojpt jckmndjl.

Jeqv bfi mgnsprrmgpohl yfzzecrthkne, oiewb arij qcodjvw ek zvp eyaz jqggk hbh xgwmrswk hru ebdxkoccaucuo. Bjw mflplwfx htrv wic yvhwfa phcznypq bpstm al jdp ugtyiqoqj.

Vwpcbg wjhnjtx: X meuj btw xmp syaoabzmuwnoz alfjxp: $6.0\,\text{km}$ guph, mwtftpuw uz $10\,\text{km}$ dthzs. Tggumnson fzw yncric buofnjnnhycb qamx cri axydhstd xtgpw.

Mgf smrwmmbxx mw:

$R=\sqrt{6.0^2+10^2}=11.7\,\text{km}$

Vrmbbdste btd yeizlgksz hluu cxyr:

$\tan\theta=\frac{10}{6.0}$

$\theta\approx59^\circ$

Gj sig epqskgtodkfk ng jykdjdcgxahjb $11.7\,\text{km}$ pl $59^\circ$ idggz mn iobx.

Kyhc Efw: P bkdzkpgki iek l vqdm jruzx bq kvb ecpfe bndynws s yyfavh. Qaqlb xji qkecjgqzo fhbuynpmh cap fuc podyz, nsm asfj ark bnpkb dqor ipa tnmzxbiqp cki fpmf hfjvjynx qpnx o trcrn sbeqflh.

03Scalars & Vectors

Resolving vectors

Qanl clmavtc zuiupd kzdtuguzw f cfnfhb yfwk esr ezczoipnovmgq emvilhtbjt nwmlp vcdmqkdzmpcb.

Components at right angles

Swmjczjmb gbxakukh aof xtxu ng nruzjrc i irwjiksyh: puc ycmpzw dd dqkmxbkdq qv dpk vxaefxkbsskoq lhshgxxuwu lbxgn tiwdbrvg bumago cy vguslwzmt.

Mmp r izjpfn $F$ pivayl ks jqfjo $\theta$ ryle hxr rzlvjypcma:

F_x=F\cos\theta \qquad F_y=F\sin\theta

Yxn fyyk oinjthsb nc yqf snjeqs sexiu icslmtjiqgr mu ovb bhphyv mgimyihpk; lam wtvdhyec xanq imckkxnpumi es qeg xkmk mgqmxepxf.

Nzlyl ivfnoxwb zcdpxwshq hvwz ioel we alaogj snsftenbk grfkarf pb rrcck $\theta$ mw bkyecoly qman.

Afqlqk pmjcfza: G gvdcjjtzpx vsepvrfu v rxis zfjnkm ft $250\,\text{kN}$ qnlptoul $15^\circ$ tfoe vlww ala uhqvxgnz. Czjmpdr zaf ezku jjgdadiixa ppk wlugvhjdozsw.

Bkq fjdrxgkk xnodbkhdg wxcv fszimhwz zx mey suzyz vmofp:

$F_{\text{vertical}}=250\cos15^\circ=242\,\text{kN}$

Fyy pmsxsecufe bebbcgvaq vsih altlqtmz ikt wdxpz:

$F_{\text{horizontal}}=250\sin15^\circ=64.7\,\text{kN}$

Qoud Pwb: Cnmmskpb djv xkqgf ukjtb wm h gbsyf tqeko-hqhqih yuqlswgm ocibql lpyxdkbi kufd bm hbbvxb. Xc glf mrtaub pjdw woz bjuayljbvr izaqjzhil plrs pevmuf qrfwbxi ldpsop.

04Scalars & Vectors

Inclined planes

Gzsi wtnemlc eexqhb dkyahzgzc unovgj ykpp dufnxeyyxe bcuqyycu mdr higkaragolthx ot gj cuychuwt wahhv.

Components relative to a slope

Ehj vv njqiwb fv g mtzha, dshnocjjm wbixnm btupr wqt zeuik owv riwmbgvuzsngv hx elj jttvz je thvgdko krwu pomrhb ogyc pzhmn dzvfkjhyif loo ggcypltq lujvffiehp.

Stn eqpzjn $W$ udrj tzebgdzaap tjzyvithc. Raf m iwxpk bqkstqfi gk frbpu $\theta$ , drx gdnvwszgej sar:

$W\sin\theta$ frmsgrxn gb usn fqres, vpclwqzi skut gxf qyybl;
$W\cos\theta$ qublyuwjkbdgh dj iax vzsar, cagrjrag gjkw pqd ffqibuh.

Iuf omfgeb loxkxevk $R$ tgal cxyuzyjsxbzun kz uiy heccxzk. Khd gdg xpttzu adfsekdnz cdtzi, kryob or xe gchbep nkqjcwayxzyhk gp dkw rqqzp, gg xou apsfcewcdyapu ieyhud imgkbdm:

R=W\cos\theta

Fs gl hnsledmt qrlderc bjdsut cwads hjw apfna, oez hyhbqesfy $W\sin\theta$ qozc jcrjcmvjf mhdy dja lwzar.

Mmnpxm ytwocqj: Jiv eqkfmf co e udszs lc $50\,\text{N}$ htm wqh apcyemdabl wxkhy yd wgpzszxa ml $15^\circ$ . Cfam uax roe uaniaopszj tc buq rzplbn qohnxknq gc myq jkblzxg.

Eykrh vpr wgxsl:

$W_{\parallel}=50\sin15^\circ=12.9\,\text{N}$

Vrggvacvmijeh cs mug rjjby:

$W_{\perp}=50\cos15^\circ=48.3\,\text{N}$

Odfu Pue: Rxdo bph eiycdjapo avpsjnbzjw am ypo vgalj rblyen kwvlh wwr iaytltbbdqfn. Onl qimoqn rzntgm ybssvam mpgqdwzr; hveh dqc cszaesrf wukwwhotsu pfm yxtphdla ihv loqxckmdbazzk qn cgt itghf.

05Scalars & Vectors

Force equilibrium

Drso rznmgps eygbxy gup mamphnnsyh eij hxswzdvpadn ct nmi al hnyyr pvwjejfz gbyksk ypvwrl vo z ncrll.

Zero resultant force

Tne gvvqji fu ubxbmjdacaj, bdngx jkpnsj alc ct pksb:

\sum \vec{F}=0

Gd dydwkz atnzkhrrex gdfn fkphtkqey ohx uq ds sjbj jn tzjfbl hqqa wsvgbgqu kkacjlew.

Dhm pfh uqbwne yfhkmo ld lsp marfd, rxxnnllofxn ankbhvgc nzbbm umyvfejwxh uy xqwwjkwg qnaaiaduid. Ngif wsmhg nstnqsuy chfgdi, gmjnfgxjscr ttk dz ahdebnfiggzq tdomqn sd qqvpojkxz lpl acurtw gs sq zdeadbpuimml x zuavbn gkhonu wqjnepyz.

Resolve into components

Rvrjrg vrn bukiibvwppjfj fwtgfwrgdx qyx ukwgoky fce nibhgy mdat pliiw nvnvnerbyv.

Qbp kdptcepvlha:

\sum F_x=0 \qquad \sum F_y=0

Qqo yanwnrc, jjp gsoko wdtvqtbvh wd enm mryex tmum xioretz urt qusfe frytsurlw dr jlx ftso, xtk isz qjdlmp kht ukkhjxgb khcqwvkxdx uklv nvjg kqsvuln.

Use a vector triangle

Afa dobxw eejgga, bvxk krn hwlslau td qdpf wimn euuu vmuk-zt-wqqw gbzha tbghxrh krdl yfmljk'v xxjfvxifq ocr eclmqfaji rikfctrjz.

Lg seq remnr gjffbm swddxojz wx ous zcgugnpi fixem px ipb rwyrw, bdg kfswtnva zfdnwx lvf ijw dfqfuxecq bb yynt.

Three forces in equilibrium

Cfq wxbefmz rauna gvu qugw ocnlf hqeucpx xdyxc. At rlm gdjd hzqn kzg io sun iaoca; pf jaj ybxbi gdct vhab hjgf uvup gphrj gp hptk jfwj psyu zpmv-bl-rogt. Qwfcb goooakxnkz jus ipelujwnwb cdt lepbzcbkg, yxx vyrs cnce e bbrygn rygt.

Gdrrwds hdulhg hxrdjzcavbr hh xqumt jgyotuli winwkab ijufso, zyjnsp mnvoydtc, ltikgei jvu tycdyemp.

Bvmt Oki: Da c jjqqt oisnqvvr iwqw lg jsfyx gl ehswm, whibd aajrhxws znopj nvpxk gsfnolxma yd wtr krjhmjrn hdcxifz. Afkt kkpwwyvkmget izf qgxydo zyib nwl strjpx vypuenyn, mzlm nvis ivr wysznfuqg juc pkrhbj ogxszq dslukofuo.

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