From 3084db1e722f7e95ae33827396744c908a5847fd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Zacharias=20Zell=C3=A9n?= Date: Mon, 26 Jan 2026 11:53:05 +0100 Subject: [PATCH] vault backup: 2026-01-26 11:53:05 --- .obsidian/workspace.json | 225 +++------------------------------------ Vektorer.md | 5 +- 2 files changed, 18 insertions(+), 212 deletions(-) diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 07e6adb..327da41 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -4,207 +4,11 @@ "type": "split", "children": [ { - "id": "277432b5491ac5a8", + "id": "9b8c02200aa5b353", "type": "tabs", "children": [ { - "id": "5d5c0fef64eecc2b", - "type": "leaf", - "state": { - "type": "markdown", - "state": { - "file": "Gräsvärde (1).md", - "mode": "source", - "source": false - }, - "icon": "lucide-file", - "title": "Gräsvärde (1)" - } - }, - { - "id": "ad6eb280b4b8718c", - "type": "leaf", - "state": { - "type": "markdown", - "state": { - "file": "Derivata.md", - "mode": "source", - 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"query": "\\text{p", + "query": "def", "matchingCase": false, "explainSearch": false, "collapseAll": false, @@ -377,23 +180,23 @@ "obsidian-git:Open Git source control": false } }, - "active": "30539ff6e6b3eb6a", + "active": "9ec7a3d1ef5d43cc", "lastOpenFiles": [ - "Funktioner.md", - "Vektorer.md", + "Primära Funktioner.md", + "ODE.md", + "Maclaurin.md", + "Komplexa tal.md", + "Integraler.md", "Gräsvärde (1).md", "Grafer.md", "Funktioner Forts.md", - "Maclaurin.md", - "ODE.md", - "Primära Funktioner.md", + "Funktioner.md", "Differential.md", - "Integraler.md", - "Definitioner.md", "Derivata.md", - "Int1.png", + "Definitioner.md", + "Vektorer.md", "Tenta Example.md", - "Komplexa tal.md", + "Int1.png", "Def_graf1.png", "TE1.png", "Trigonometri.md", diff --git a/Vektorer.md b/Vektorer.md index 4377952..9f2227d 100644 --- a/Vektorer.md +++ b/Vektorer.md @@ -1,10 +1,13 @@ - **DEF** - *I en rätviklig rektangle stämmer $\overrightarrow{AC}=\left(\overrightarrow{AB}+\overrightarrow{AD}\right)$* - $\overrightarrow{u}=\left(1,2,3\right)=\left(\begin{aligned}1\\2\\3\end{aligned}\right)=\left[\begin{aligned}1\\2\\3\end{aligned}\right]$ + - *$\mid\mid{V}\mid\mid$ Är längden av $V$* - **Exemple** - $$\begin{align}\text{Rektangeln }A,\;B,\;C,\;D\;\text{. Låt }E\text{ Vara punkten som delar diagonalen }\overline{AC}:\text{förhållandet }1:3\\\left(\text{dvs: }\overline{AE}:\overline{EC}=1:3\right)\\\text{Benämna }\overrightarrow{a}=\overrightarrow{AB},\overrightarrow{h}=\overrightarrow{AD}\text{ Uttryc vektor }\overrightarrow{c}=\overrightarrow{EC}\text{ i termer av }\overrightarrow{a}\text{ och }\overrightarrow{h}\\\\\text{Vart ligger punkten }E\:\text{? Hur kan vi uttrycka }\overrightarrow{c}\text{ med hjälp av }\overrightarrow{a}\text{ och }\overrightarrow{h}\:\text{?}\\\overrightarrow{c}=\overrightarrow{EC}=\frac34\overrightarrow{AC}=\frac34\left(\overrightarrow{AB}+\overrightarrow{BC}\right)=\frac34\left(\overrightarrow{a}+\overrightarrow{h}\right)=\frac34\overrightarrow{a}+\frac34\overrightarrow{h}\end{align}$$ - **Koordinatrummet $\mathbb{R}^m$** - *Mängden $\mathbb{R}^m\;\left(\text{där }m\in\mathbb{N}\right)$ består av koordinattpunkter av längden $m$ vars element är reella tal. Som skalärer tas $\mathbb{R}\text{(vanliga reela tal)}$* - **Hur funkar $+$ och $\times$** - $$\begin{align}\overrightarrow{u}=\left(u_1,u_2,u_3,\dots,u_m\right)\in\mathbb{R}^m\\\overrightarrow{v}=\left(v_1,v_2,\dots,y_m\right)\in\mathbb{R}^m\\\lambda\in\mathbb{R}\\\\\overrightarrow{u}+\overrightarrow{v}=\left(u_1+v_1,u_2+v_2,\dots,\lambda u_m\right)\\\text{OBS: }\left(1,2\right)+\left(3,4,5\right)\Rightarrow\text{Inte Definierat}\\\\\overrightarrow{u}\times\overrightarrow{v}=\left(u_1v_1,u_2v_2\dots,u_mv_m\right)\\\begin{aligned}\overrightarrow{u}=\left(1,2,0\right)\\\overrightarrow{v}=\left(0,0,-2\right)\end{aligned}\Rightarrow\overrightarrow{u}\times\overrightarrow{v}=\left(1\times0,2\times0,0\times\left(-2\right)\right)=\left(0,0,0\right)\\\text{Man kan i normala fall inte multiplecera vektorer!}\end{align}$$ - - \ No newline at end of file + - $$\overrightarrow{u},\overrightarrow{v}\in\mathbb{R}^3\Rightarrow\overrightarrow{n},\overrightarrow{n}\in\mathbb{R}^3$$ + - **Sats**: *Låt $\overrightarrow{m},\:\overrightarrow{n}\in\mathbb{R}^3$. Då gäller det att: $\mid\mid\overrightarrow{m}\times\overrightarrow{n}\mid\mid=\mid\mid\overrightarrow{m}\mid\mid\times\mid\mid\overrightarrow{n}\mid\mid\times\sin(\theta)$. (Där $\theta$ är vinkeln mellan $\overrightarrow{m}$ och $\overrightarrow{n}$). (Jämför: $<\overrightarrow{u},\:\overrightarrow{v}>=\mid\mid\overrightarrow{u}\mid\mid\times\mid\mid\overrightarrow{v}\mid\mid\times\cos(\theta)$)* + - **Prof**: *Vi börjar med: $$\begin{aligned}\mid\mid\overrightarrow{u}\times\overrightarrow{v}\mid\mid^2=<\overrightarrow{u}\times\overrightarrow{v},\:\overrightarrow{u}\times\overrightarrow{v}>\stackrel{\text{(I)}}{=}<\overrightarrow{u},\;\overrightarrow{v}\times\left(\overrightarrow{u}\times\overrightarrow{v}\right)>\stackrel{\text{(II)}}{=}<\overrightarrow{u},\;<\overrightarrow{v},\;\overrightarrow{v}>\overrightarrow{u}-<\overrightarrow{v},\;\overrightarrow{u}>\overrightarrow{v}>\\=<\overrightarrow{u},\;<\overrightarrow{v},\;\overrightarrow{v}>\overrightarrow{u}>-<\overrightarrow{u},\;<\overrightarrow{v},\;\overrightarrow{u}>\overrightarrow{v}>\\=<\overrightarrow{v},\;\overrightarrow{v}><\overrightarrow{u},\;\overrightarrow{u}>-<\overrightarrow{u},\;\overrightarrow{v}><\overrightarrow{v},\;\overrightarrow{u}>\\=\mid\mid\overrightarrow{u}\mid\mid^2\times\mid\mid\overrightarrow{v}\mid\mid^2-\left(\mid\mid\overrightarrow{u}\mid\mid\times\mid\mid\overrightarrow{v}\mid\mid\times\cos(\theta)\right)^2\\=\mid\mid\overrightarrow{u}\mid\mid^2\times\mid\mid\overrightarrow{v}\mid\mid^2\times\left(1-\cos^2(\theta)\right)\\=\mid\mid\overrightarrow{u}\mid\mid^2\times\mid\mid\overrightarrow{v}\mid\mid^2\times\sin^2(\theta)\end{aligned}$$* \ No newline at end of file