5. Vektorer
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Kapitel 5
5.
Vektorer
Vektorer är matematiska objekt som beskriver både storlek och riktning. De används inom fysik och matematik för att representera storheter som hastighet, acceleration och kraft. Vektorer kan beskrivas med koordinater, och deras längd kan beräknas med Pythagoras sats. De kan också representeras grafiskt som pilar. Vektorer kan parallellförflyttas utan att ändra sina egenskaper, och de kan beskrivas i både vågrät och lodrät form. Vektorer är skiljer sig från skalärer, vilka har storlek men ingen riktning. Ett exempel på en vektor är vindens fart och riktning.
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Vektorer
Sida av 12
I den här lektionen går vi igenom följande begrepp:
- Vektor
- Parallellavektorer
- Motsatta vektorer
- Koordinatform för vektor
- Parallellförflyttning av vektor
- Komposant
- Längden av en vektor
Förkunskaper
Teori
Vektor
En vektor beskriver en storhet som har både storlek och riktning. Storheter som kan beskrivas med vektorer är t.ex. hastighet, acceleration och kraft. Den vanligaste notationen för vektorer är att sätta en pil eller ett streck över en bokstav, exempelvis
vellervˉ.
Grafiskt brukar en vektor representeras av en pil, där pilens längd motsvarar vektorns storlek och pilhuvudet visar riktningen. Drar man en vektor mellan två namngivna punkter, t.ex. startpunkten A och slutpunkten B, brukar man namnge den AB, och om den riktas åt andra hållet får den namnet BA. Teori
Parallella vektorer
Två vektorer, v1 och v2, sägs vara parallella om de har samma riktning eller om de har motsatta riktningar.
Teori
Motsatta vektorer
Två vektorer, v1 och v2, sägs vara motsatta om de har samma längd men motsatta riktningar.
v2=−v1
Följande applet kan användas för att visualisera en vektor och dess motsatta vektor. 
Exempel
Beskriver ett par vektorer
Betrakta följande par av vektorer.
a Är vektorerna parallella?
{"type":"choice","form":{"alts":["Ja","Nej"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
b Är vektorerna motsatta?
{"type":"choice","form":{"alts":["Ja","Nej"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
Ledtråd
a Parallella vektorer har samma eller motsatt riktning.
b Motsatta vektorer har samma längd och motsatt riktning.
Lösning
a Parallella vektorer har samma riktning eller motsatt riktning. Detta innebär att om de placeras tillsammans, kommer de att ligga på samma linje.
Vektorerna har motsatt riktning. Detta innebär att de är parallella.
b Motsatta vektorer har samma längd och motsatt riktning. Det konstaterades i del A att de givna vektorerna har motsatt riktning. Dock kan det ses, genom att placera dem bredvid varandra, att de inte har samma längd.
Eftersom vektorerna inte har samma längd, är de inte motsatta.
Teori
Koordinatform för vektor
Vektorer brukar beskrivas i koordinatform, där koordinaterna anger förändringen i x- och y-led. För en utritad vektor kan man bestämma koordinaterna genom att beräkna skillnaden i x- och y-led mellan start- och slutpunkten.
Till skillnad från punkter på koordinatform, som anger en specifik position i koordinatsystemet, anger vektorer förändring, och är alltså inte bundna till en viss position.
Exempel
Skriv vektorerna på koordinatform
Skriv vektorerna u och v på koordinatform.
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Ledtråd
Använd rutnätet för att hitta den horisontella och den vertikala förskjutningen för varje vektor.
Lösning
För att skriva vektorerna på koordinatform mäter vi förändringen i x- och y-led mellan start- och slutpunkterna. För v noterar vi att slutpunkten finns till vänster om startpunkten, vilket ger en negativ förändring i x-led.
Skillnaden är 4 längdenheter i x-led och 2 i y-led. Det innebär att koordinatformen för u är (4,2). Skillnaden i x-led är −3 längdenheter och 1 i y-led, så koordinatformen för v är (−3,1).
Teori
Parallellförflyttning av vektor
När en vektor flyttas utan att vridas eller ändra längd sägs den ha parallellförflyttas. Vektorn hamnar i en annan position i koordinatsystemet men beskrivs av samma koordinater eftersom ändringen i x- och y-led inte beror på var vektorn är. Exempelvis beskrivs alla vektorer i figuren av v=(3,3).
Parallellförflyttas vektorn så att startpunkten hamnar i origo kommer slutpunkten att få samma koordinater som själva vektorn.
Teori
Komposant
En vektor kan alltid delas upp i två eller flera delvektorer
som anger förändringar i olika riktningar. Dessa delvektorer kallas komposanter. Oftast delar man upp dem i en vågrät x-komposant och en lodrät y-komposant, vars längder ges av vektorns koordinater. I figuren har vektorn v=(6,4) delats upp i komposanterna vx=(6,0) och vy=(0,4).
Exempel
Dela upp vektor i komposanter
Dela upp vektorn v i komposanter.
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{"type":"text","form":{"type":"point2d","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false,"decimal":false,"function":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.000108em;vertical-align:-0.286108em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15139200000000003em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.03588em;\">y<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.2355em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text1":"0","text2":"-8"}}
Ledtråd
Använd rutnätet för att hitta den horisontella och den vertikala förskjutningen för varje vektor. Skriv en vektor för var och en av dessa förskjutningar.
Lösning
Från startpunkt till slutpunkt ser vi att v ändras med 7 steg åt vänster och 8 steg ner i rutnätet vilket ger oss dess koordinatform: (−7,−8). Vi kan dela upp denna vektor i en vågrät x-komposant och en lodrät y-komposant som nedan.
Den vågräta komposanten visar hur vektorn ändrats i x-led, dvs. −7 steg så yx har koordinatformen (−7,0). På samma sätt visar den lodräta komposanten vektorns förändring i y-led, dvs. −8 steg så koordinatformen för vy blir (0,−8).
Teori
Längden av en vektor
Längden av en vektor v brukar skrivas ∣v∣, vilket utläses normen eller absolutbeloppet av v. Längden på vågräta och lodräta vektorer kan avläsas direkt i koordinatsystemet, men mer generellt går det att använda Pythagoras sats för att beräkna längden.
∣(a,b)∣2=a2+b2.
Dras kvadratroten ur båda led ger detta den generella formeln för längden av en vektor. Eftersom längder alltid är positiva bortser man från den negativa roten. ∣(a,b)∣=a2+b2
Exempel
Vad är vektorns längd?
Vad är längden av vektorerna u och v?
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Ledtråd
Använd formeln för en vektors längd.
Lösning
Vi går igenom en vektor i taget.
Vektor u
Vektorn u är lodrät och längden av en sådan vektor kan bestämmas genom att räkna hur många rutor den sträcker sig. Vi ser att den löper fem rutor uppåt så längden av vektorn blir alltså 5 le.
Vektor v
Denna vektor är varken lodrät eller vågrät så för att bestämma dess längd sätter vi in koordinatformen i formeln∣(a,b)∣=a2+b2.
Koordinatformen anger förändringen i x- och y-led mellan vektorns start- och slutpunkt. Från rutnätet ser vi att förändringen i x- och y-led är 4 och 3 så vektorn är (4,3).
∣(a,b)∣=a2+b2
SubstituteII
a=4 och b=3
∣(4,3)∣=42+32
CalcPow
Beräkna potens
∣(4,3)∣=16+9
AddTerms
Addera termer
∣(4,3)∣=25
CalcRoot
Beräkna rot
∣(4,3)∣=5
Längden på vektorn v är också 5 le.
Vektorer
Övningar
A.B.C.D.E.F.G.Temperaturen i klassrummet.Volymen juice i ett glas.Din massa.En cyklists hastighet.Kraften du anva¨nder fo¨r att o¨ppna en do¨rr.Din position i fo¨rha˚llande till mataffa¨ren.Avsta˚ndet till skolan.
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security
report
code
security
report
code
Alla storheter har en storlek, men endast vektorer har en riktning. Vi undersöker dem i ordning.
A till C
Varken temperatur, volym eller massa kan ha riktning. Det kan inte vara 22 ^(∘)C åt vänster
, juicens volym kan inte vara 2dl åt vänster
och din massa är inte 75kg åt vänster
. Alla tre är skalärer.
D till E
En hastighet har däremot en riktning. Du kan säga cyklistens hastighet är 30.km /h. åt vänster
. Det är därför en vektor. På liknande sätt spelar det stor roll om du drar med en kraft uppåt eller nedåt i dörrhandtaget. Därför är även kraft en vektor.
F till G
Position och avstånd kan tyckas vara samma sak, men det finns en tydlig skillnad. En apelsin kan ha positionen 1m framför dig, 1m till höger om dig osv. Oavsett var apelsinen befinner sig är avståndet till apelsinen alltid detsamma, 1m. En position anges som hur långt och åt vilket håll i förhållande till något, och är alltså en vektor. Avståndet anger däremot bara hur långt
, och är därför en skalär. Vi sammanfattar svaren i en tabell.
| Storhet | Skalär/Vektor | |
|---|---|---|
| A | Temperatur | Skalär |
| B | Volym | Skalär |
| C | Massa | Skalär |
| D | Hastighet | Vektor |
| E | Kraft | Vektor |
| F | Position | Vektor |
| G | Avstånd | Skalär |
I lösningen till deluppgift A har vi funnit att alternativen A, B, C och G skalärer.
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Ange vektorn på koordinatform.
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{"type":"text","form":{"type":"point2d","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false,"decimal":false,"function":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.20772em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text1":"-6","text2":"2"}}
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Vi bestämmer skillnaden i x- och y-led mellan start- och slutpunkterna för vektorn.
Dessa skillnader, dvs. Δ x och Δ y, ger oss x- och y-koordinaterna för vektorn.
| Startpunkt | Slutpunkt | Δ x | Δ y |
|---|---|---|---|
| (- 10,2) | (- 4,6) | -4 -(- 10)=6 | 6-2=4 |
Vektorn kan alltså skrivas v=(6,4).
Återigen bestämmer vi skillnaden i x- respektive y-led mellan vektorns start- och slutpunkter.
Vi får nu x- och y-koordinaten för vektorn genom skillnaderna Δ x och Δ y.
| Startpunkt | Slutpunkt | Δ x | Δ y |
|---|---|---|---|
| (8,- 6) | (2, - 4) | 2-8=-6 | - 4-(- 6)=2 |
Vi kan alltså skriva vektorn som u=(-6,2).
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Vilka vektorer är likadana?
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style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.2355em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">v<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.20772em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.20772em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">v<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.20772em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 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preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.20772em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.2355em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.20772em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.02691em;\">w<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.02691em;\">w<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.2355em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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Två vektorer är identiska om de har samma storlek och riktning. Vi ser att alla vektorer är lika långa, så de har alla samma storlek. Men vilka har samma riktning? Vi kan parallellförflytta dem för att lättare se det.
Nu ser vi att v och a är parallella, så de är lika. u och w lutar på samma sätt, men har motsatt riktning (även kallat antiparallella). De beskriver alltså inte samma vektor. Sammanfattningsvis gäller alltså v=a.
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I diagrammet finns tre vektorer.
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{"type":"text","form":{"type":"point2d","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false,"decimal":false,"function":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">v<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.20772em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text1":"5","text2":"4"}}
{"type":"text","form":{"type":"point2d","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false,"decimal":false,"function":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.02691em;\">w<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text1":"7","text2":"-6"}}
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För att skriva en vektor på koordinatform bestämmer vi skillnaden i x- och y-led mellan start- och slutpunkt. Vektorern u har startpunkt i (0,0), dvs. origo, så dess koordinatform ges av slutpunktens koordinater.
Nu kan vektorns koordinatform skrivas. u=(- 9,2)
Men v börjar inte i origo, så för den måste vi bestämma skillnaden i x- och y-led.
Med denna information kan vi skriva vektorns koordinatform. v=(5,4)
Precis som i deluppgift A finner vi att w har startpunkt i (0,0). Därför bestäms dess koordinatform av slutpunktens koordinater.
Vektor w kan alltså i koordinatform skrivas som w=(7,- 6).
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Kan följande beskrivas med en vektor eller en skalär?
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{"type":"choice","form":{"alts":["Vektor","Skal\u00e4r"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
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En vektor beskriver både storlek och riktning. Kostnaden för en bil har visserligen en storlek, exempelvis 50 000 kr, men saknar riktning, Detta är alltså en skalär.
Vinden har både fart (dvs. storlek) och blåser i en riktning. Detta är därför en vektor.
Sveriges befolkning har en storlek, ca 9 miljoner men har befolkningen en riktning? Nej, en befolkning har ingen riktning så det måste vara en skalär.
Ett flygplan åker i en viss riktning. Flygresan har alltså både en storlek (farten) och riktning. Det är en vektor.
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Bestäm koordinaterna för vektorerna a, b och c. Varje ruta är 1 l.e.
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9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9774399999999999em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">b<\/span><\/span><\/span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.2355em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">c<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.17994em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>"}],[{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">4<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">4<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2],[0,1,2]]}
{"type":"pair","form":{"alts":[[{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.2355em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9774399999999999em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9774399999999999em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">b<\/span><\/span><\/span><span style=\"top:-3.26344em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.2355em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.714em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">c<\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.17994em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 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class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">4<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">1<\/span><span 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Koordinaterna för en vektor anger vektorns förändring i x- och y-led. Om vi börjar med att titta på a, ser vi att den rör sig 4 rutor åt höger i x-led och 1 ruta uppåt i y-led.
Vektor a:s koordinater blir alltså a=(4,1). Nu gör vi på samma sätt med de andra vektorerna. Kom ihåg att om vektorn går nedåt eller åt vänster blir koordinaten negativ.
Koordinaterna blir alltså b=(-2,-4) och c=(-2,1).
Vi ser att vi nu har vi samma vektorer igen, fast inplacerade i ett koordinatsystem. Eftersom en vektors koordinater endast beror på förändringen i 'x- och 'y-led, och inte på var vektorn är placerad, är koordinaterna identiska med dem i förra deluppgiften.
Vektorerna har alltså koordinaterna a=(4,1), b=(-2,-4) och c=(-2,1).
Vi kan även lösa uppgiften genom att parallellförflytta vektorerna till origo och avläsa koordinaterna.
Vi ser att vektorernas slutpunkter hamnar i (4,1), (-2,-4) och (-2,1). Vektorernas koordinater blir då samma, a=(4,1), b=(-2,-4) och c=(-2,1).
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Rita vektorn och identifiera sedan vilket av diagrammen som representerar den.
v=(4,2)
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
w=(5,−3)
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
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Vektorn v=(4,2) representeras av en pil som går 4 steg åt höger på x-axeln och 2 steg uppåt på y-axeln.
Detta motsvarar alternativ C.
Vektorn w=(5,-3) representeras av en pil som går 5 steg åt höger på x-axeln och 3 steg nedåt på y-axeln.
Detta motsvarar alternativ A.
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Beräkna längden av vektorn. Avrunda till två decimalers noggrannhet.
v=(−4,7)
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u=(−8,−5)
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Vi använder formeln för beräkning av en vektors längd. En av koordinaterna är negativ, men formeln fungerar oavsett.
Längden på vektorn är alltså sqrt(65) le. vilket är ungefär 8.06 le.
Vi gör på samma sätt och sätter in vektorns koordinater i längdformeln.
Längden blir sqrt(89)≈ 9,43 le.
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I nedanstående koordinatsystem har vektorn v ritats.
Rita komposanter vx och vy för vektorn, dvs. vektorns vågräta och lodräta komposant.
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.036108em;vertical-align:-0.286108em;\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15139200000000003em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.03588em;\">y<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.2355em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["8"]}}
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{"type":"text","form":{"type":"point2d","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false,"decimal":false,"function":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.000108em;vertical-align:-0.286108em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15139200000000003em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right:0.03588em;\">y<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"accent-body\" style=\"left:-0.2355em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text1":"0","text2":"8"}}
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Komposanten i y-led börjar i startpunkten till v och går längs med y-axeln tills den når samma y-koordinat som v. Komposanten i x-led drar vi längs med x-axeln tills den når samma x-koordinat som v.
Lod- och vågräta vektorer i ett koordinatsystem är enkla att bestämma längden på, då det bara är att räkna hur många steg de går i höjd- eller sidled.
Den vågräta komposanten v_x är 6 l.e. och den lodräta komposanten v_y är 8 l.e.
Eftersom komposanterna börjar i origo kan deras koordinatform utläsas av vektorns slutpunkt. Den vågräta vektorn slutar i punkten (- 6, 0) och den lodräta slutar i (0,8) så deras koordinater blir
v_x=(- 6,0) och v_y=(0,8).
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