MATHHX A
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12.2 Koordinater for vektorer
Indtil videre har der været en masse tegning og ikke så meget regning. Det skal der laves om på! Men skal vi regne med vektorer, er vi nødt til at beskrive dem på en måde, så man kan regne på dem. Det gør vi med koordinater.
-
Definition 12.2.1
En vektor \(\vec {a}\) tildeles koordinater som vist på tegningen:
På tegningen er \(a_1\) og \(a_2\) positive. De kan også være negative, så går vi bare i den modsatte retning.
Vi skriver \(\vec {a}=\begin {pmatrix}a_1\\ a_2\end {pmatrix}\)
-
Eksempel 12.2.1
Betragt vektorerne
Vi aflæser at
\[\vec {a}=\begin {pmatrix}1\\ 0\end {pmatrix},\quad \vec {b}=\begin {pmatrix}-1\\ -2\end {pmatrix}\quad \text {og}\quad \vec {c}=\begin {pmatrix}0\\ 2\end {pmatrix},\quad \]
Øvelse 12.2.1 (Svær)
Betragt vektorerne:
\[\vec {a}=\begin {pmatrix}2\\ 1\end {pmatrix},\quad \vec {b}=\begin {pmatrix}1\\ -1\end {pmatrix}\quad \vec {c}=\begin {pmatrix}-1\\ -1\end {pmatrix}\quad \text {og}\quad \vec {d}=\begin
{pmatrix}0\\ 0\end {pmatrix}\]
Løsning 12.2.1
-
a)
Regning med koordinater fungerer helt som forventet:
-
Sætning 12.2.1
Lad
\[\vec {a}=\begin {pmatrix}a_1\\ a_2\end {pmatrix}\quad \text {og}\quad \vec {b}=\begin {pmatrix}b_1\\ b_2\end {pmatrix}\]
Så er
\[\vec {a}+\vec {b}=\begin {pmatrix}a_1+b_1\\ a_2+b_2\end {pmatrix},\quad \vec {a}-\vec {b}=\begin {pmatrix}a_1-b_1\\ a_2-b_2\end {pmatrix} \quad \text {og}\quad t\vec {a}=\begin
{pmatrix}t\cdot a_1\\ t\cdot a_2\end {pmatrix}\]
Øvelse 12.2.2 (Svær)
Du skal nu tjekke påstanden i ovenstående sætning
-
a) Tegn to vilkårlige vektorer og tegn sum, differens, og konstant gange vektor. Indtegn koordinater og tjek, at sætningens påstand er rigtig.
Øvelse 12.2.3
Betragt vektorerne:
\[\vec {a}=\begin {pmatrix}2\\ 1\end {pmatrix}\quad \text {og}\quad \vec {b}=\begin {pmatrix}1\\ -1\end {pmatrix}\]
Regn:
-
a) \(\vec {a}+\vec {b}\)
-
b) \(\vec {a}-\vec {b}\)
-
c) \(2\vec {a}\)
Løsning 12.2.3
-
a) \(\vec {a}+\vec {b}=\begin {pmatrix}3\\ 0\end {pmatrix}\)
-
b) \(\vec {a}-\vec {b}=\begin {pmatrix}1\\ 2\end {pmatrix}\)
-
c) \(2\vec {a}=\begin {pmatrix}4\\ 2\end {pmatrix}\)
Længden af en vektor
Længden af en vektor \(\vec {a}\) betegnes med \(|\vec {a}|\).
Øvelse 12.2.6
Betragt vektorerne:
-
a) Bestem længden af \(\vec {a}\).
-
b) Bestem længden af \(\vec {b}\).
-
c) Bestem længden af \(\vec {c}\).
Øvelse 12.2.7
Lad \(\vec {F}=\vectwo {3}{2}\). En repræsentant for \(\vec {F}\) er tegnet ind i et koordinatsystem:
Lad \(\theta \) betegne vinklen mellem \(F\) og \(x\)-aksen. Du skal nu beregne \(\theta \). Jeg hjælper lidt.
-
a) Hvad mener jeg mon med vinklen mellem \(F\) og x-aksen? Lav en tegning, hvor man kan se den.
-
b) Indtegn en retvinklet trekant som du kan bruge til at bestemme \(\theta \).
-
c) Du kender længden af to trekantens sider. Hvilke?
-
d) Beregn \(\theta \).
Løsning 12.2.7
-
a) Den stiplede linje er parallel med \(x\)-aksen
-
b)
-
c) Koordinaterne for \(\vec {F}\) udgør trekantens kateter.
-
d) \(\theta =33{,}69\degree \)
Øvelse 12.2.8
Lad \(\vec {a}\) være en egentlig vektor. Antag at vinklen mellem \(\vec {a}\) og \(x\)-aksen er \(53{,}13\degree \) og at \(|\vec {a}|=5\).
Komposanter
Forstil dig du har en glat klods på en glat overflade. Forstil dig at der bliver trukket i klodsen i to retninger. Vi repræsenterer kræfterne med to vektorer \(\vec {F_1}\) og \(\vec {F_2}\):
I fysik lærer man, at klodsen vil nu bevæge sig som der kun var en kraft, nemlig summen af de to kræfter \(F_\text {res}\), som vist her:
Dette princip kan man udnytte når man analysere kræfter i fysik. Lad os se på et eksempel, hvor vi nu gang placerer klodsen på et glat skråplan:
Det ses ,at der kun er én kraft som virker på klodsen, nemlig tyngdekraften. Men det er umiddelbart svært at finde ud af, hvordan klodsen bevæger sig, fordi den bevægelse afhænger af skråplanets hældning. Jo stejlere skråplanet
er, jo hurtigere vil klodsen glide ned. Vi løser problemet ved at beskrive \(\vec {F_t}\) som summen af to kræfter. En der peger langs skråplanet og en, som peger vinkelret ned på skråplanet:
De to vektorer \(\vec {F_1}\) og \(\vec {F_2}\) kaldes komposanter til \(\vec {F_t}\). Komposanter til en vektor \(\vec {v}\) er altså vektorer \(\vec {v_1}\), \(\vec {v_2}\), som opfylder, at
\[\vec {v}=\vec {v_1}+\vec {v_2}\]
Hvordan man regner klodsens bevægelse ud fra komposanterne \(\vec {F_1}\) og \(\vec {F_2}\) er mere fysik end matematik, så det vil jeg overlade til Mogens (eller hvad jeres fysiklærer nu hedder). I stedet skal I regne en
øvelse, hvor I får lov at bestemme længderne af de to komposanter.
Øvelse 12.2.9
Betragt klodsen fra før. Lad os kalde hældningsvinklen for \(\theta \).
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a) Gør rede for, at den brune vinkel vinkel er \(\theta \).
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b) Gør rede for, at den røde vinkel er \(\theta \).
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c) Gør rede for, at \(|\vec {F_2}| =|\vec {F_t}|\cdot \cos (\theta )\)
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d) Gør rede for, at \(|\vec {F_1}| =|\vec {F_t}|\cdot \sin (\theta )\)
Løsning 12.2.9
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a) Se øvelse 11.1.5
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b) Se f.eks. eksempel 11.1.1
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c) Brug reglen om cosinus i en retvinklet trekant på den brune vinkel.
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d) Brug reglen om sinus i en retvinklet trekant på den røde vinkel.