MATHHX A
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\(\newcommand {\watt }{\mathrm {W}}\)
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\(\newcommand {\THz }{\tera \hertz }\)
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\(\newcommand {\MN }{\mega \newton }\)
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7.1 Introduktion til vektorer
-
Definition 7.1.1
En vektor er en størrelse, som har både en længde og en retning. En vektor tegnes som en
pil, hvor pilens retning er vektorens retning, og pilens længde er vektorens længde.
Vektoren på tegningen læses ”vektor a”. Der findes en særlig vektor som har en længde på nul, men ikke nogen retning. Den kaldes nulvektoren, betegnes med \(\vec {0}\) og tegnes som en prik:
En vektor som ikke er nulvektoren kaldes en egentlig vektor.
Længen af \(\vec {a}\) betegnes med \(|\vec {a}|\).
Vi tegner nogle gange vektorer ind i koordinatsystemer.
Her kunne det godt se ud som jeg har tegnet to forskellige vektorer, men fordi \(\vec {a}\) og \(\vec {b}\) har samme størrelse og retning, så er de ens. Altså \(\vec {a}=\vec {b}\). Man siger at de to pile er to forskellige
repræsentanter for den samme vektor.
Øvelse 7.1.1
Betragt vektorerne
Regning med vektorer
Lad os se på hvordan vi regner med vektorer. Vi starter med addition.
Øvelse 7.1.2
Betragt vektorerne
Tegn vektorerne
-
a) \(\vec {a}+\vec {b}\)
-
b) \(\vec {a}+\vec {c}\)
-
c) \(\vec {b}+\vec {c}\)
Ligesom vi har negative tal har vi negative vektorer. De er givet ved følgende oplagte definition.
Øvelse 7.1.3
Her er en vektor:
Løsning 7.1.3
-
a)
Når vi nu har negative vektorer er der en oplagt måde at definere subtraktion.
Øvelse 7.1.4
Betragt vektorerne
Tegn vektorerne
-
a) \(\vec {a}-\vec {b}\)
-
b) \(\vec {b}-\vec {a}\)
-
c) \(\vec {b}-\vec {c}\)
Øvelse 7.1.5
Betragt vektorerne \(\vec {a}\) og \(\vec {b}\):
Man kan argumentere for at \(\vec {a}-\vec {b}\) går fra spidsen af \(\vec {b}\) til spidsen af \(\vec {a}\) som vist her:
Øvelse 7.1.6
Nogle bøger definere sum og differens af to vektorer lidt anderledes. Givet to vektorer
så defineres \(\textcolor {TextGreen}{\vec {a}+\vec {b}}\) og \(\textcolor {red}{\vec {a}-\vec {b}}\) ved at tegne et parallelogram:
Løsning 7.1.6
Nu ligger \(\textcolor {blue}{\vec {b}}\) i forlængelse af \(\vec {a}\) og derfor må den grønne vektor være \(\vec {a}+\vec {b}\).
Det var plus og minus, så mangler vi bare gange og dividere, right? Nej man kan ikke gange eller dividere to vektorer. Men man kan gange en vektor med et tal.
Øvelse 7.1.7
Her er en vektor:
Løsning 7.1.7
-
a)
Vi har brug for nogle regneregler for at gøre det nemmere at regne mere komplicerede udtryk.
Regnereglerne afspejler dem som vi kender for de reelle tal, og vi bruger sætningen til at sikre os at vi kan regne med vektorer som om de var tal – altså lige bortset fra at der ikke findes multiplikation/division mellem vektorer.
Øvelse 7.1.8
Betragt vektorerne
Reducer og tegn følgende vektorer
-
a) \(\vec {u}=2\vec {a}+\vec {a}-\vec {a}-3\vec {a}\)
-
b) \(\vec {v}=\vec {b}+\vec {a}-2\vec {a}-\vec {b}+\vec {0}+\vec {a}\)
-
c) \(\vec {w}=\vec {a} + 2( \vec {b}+\vec {c})- \vec {c}\)
Øvelse 7.1.9
Forklar følgende begreber:
Løsning 7.1.9
-
a) En vektor er et objekt med en størrelse og en retning
-
b) Nulvektoren er en vektor med længde på nul. Den tegnes som en prik.
-
c) En egentlig vektor er en vektor som ikke er nulvektoren, dvs. den en længde der er større end nul.
-
d) En repræsentant for en vektor er når vi tegner vektoren en i et koordinatsystem. Vi kan lægge den alle steder og derfor er der uendelige mange
repræsentanter for en givet vektor.