vault backup: 2026-01-20 09:26:43

.obsidian/workspace.json

@@ -195,16 +195,30 @@
             "state": {
               "type": "markdown",
               "state": {
-                "file": "Funktioner Forts.md",
+                "file": "Funktioner.md",
                 "mode": "source",
                 "source": false
               },
               "icon": "lucide-file",
-              "title": "Funktioner Forts"
+              "title": "Funktioner"
+            }
+          },
+          {
+            "id": "30539ff6e6b3eb6a",
+            "type": "leaf",
+            "state": {
+              "type": "markdown",
+              "state": {
+                "file": "Vektorer.md",
+                "mode": "source",
+                "source": false
+              },
+              "icon": "lucide-file",
+              "title": "Vektorer"
             }
           }
         ],
-        "currentTab": 13
+        "currentTab": 14
       }
     ],
     "direction": "vertical"
@@ -261,8 +275,7 @@
       }
     ],
     "direction": "horizontal",
-    "width": 300,
-    "collapsed": true
+    "width": 300
   },
   "right": {
     "id": "b700e0cd0f882a5c",
@@ -278,7 +291,7 @@
             "state": {
               "type": "backlink",
               "state": {
-                "file": "Grafer.md",
+                "file": "Funktioner Forts.md",
                 "collapseAll": false,
                 "extraContext": false,
                 "sortOrder": "alphabetical",
@@ -288,7 +301,7 @@
                 "unlinkedCollapsed": true
               },
               "icon": "links-coming-in",
-              "title": "Backlinks for Grafer"
+              "title": "Backlinks for Funktioner Forts"
             }
           },
           {
@@ -326,13 +339,13 @@
             "state": {
               "type": "outline",
               "state": {
-                "file": "Komplexa tal.md",
+                "file": "Funktioner Forts.md",
                 "followCursor": false,
                 "showSearch": false,
                 "searchQuery": ""
               },
               "icon": "lucide-list",
-              "title": "Outline of Komplexa tal"
+              "title": "Outline of Funktioner Forts"
             }
           },
           {
@@ -364,19 +377,20 @@
       "obsidian-git:Open Git source control": false
     }
   },
-  "active": "76c8d943958d45bf",
+  "active": "30539ff6e6b3eb6a",
   "lastOpenFiles": [
+    "Funktioner.md",
+    "Vektorer.md",
+    "Gräsvärde (1).md",
+    "Grafer.md",
+    "Funktioner Forts.md",
     "Maclaurin.md",
     "ODE.md",
-    "Funktioner.md",
-    "Gräsvärde (1).md",
     "Primära Funktioner.md",
     "Differential.md",
     "Integraler.md",
-    "Grafer.md",
     "Definitioner.md",
     "Derivata.md",
-    "Funktioner Forts.md",
     "Int1.png",
     "Tenta Example.md",
     "Komplexa tal.md",

Funktioner.md

@@ -1,4 +1,4 @@
-- Talmängder:
+k- Talmängder:
 	- De natuliga talen: $\mathbb{N} = \{0, 1, 2, ...\}$
 	- De hela talen: $\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$
 	- De rationella talen: $\mathbb{Q} = \{\frac{p}{q}: p, q \in \mathbb{Z}, q\neq0\}$

Vektorer.md

@@ -0,0 +1,10 @@
+- **DEF**
+	- *I en rätviklig rektangle stämmer $\overrightarrow{AC}=\left(\overrightarrow{AB}+\overrightarrow{AD}\right)$*
+	- $\overrightarrow{u}=\left(1,2,3\right)=\left(\begin{aligned}1\\2\\3\end{aligned}\right)=\left[\begin{aligned}1\\2\\3\end{aligned}\right]$
+- **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 $\times$**
+		- $$\begin{align}\overrightarrow{u}=\left(u_1,u_2,u_3,\dots,u_m\right)\in\mathbb{R}^m\\\overrightarrow{v}=\left(v_1,v_2,\dots,y_m\right)\in\mathbb{R}^m\\\lambda\in\mathbb{R}\\\\\overrightarrow{u}+\overrightarrow{v}=\left(u_1+v_1,u_2+v_2,\dots,\lambda u_m\right)\\\text{OBS: }\left(1,2\right)+\left(3,4,5\right)\Rightarrow\text{Inte Definierat}\\\\\overrightarrow{u}\times\overrightarrow{v}=\left(u_1v_1,u_2v_2\dots,u_mv_m\right)\\\begin{aligned}\overrightarrow{u}=\left(1,2,0\right)\\\overrightarrow{v}=\left(0,0,-2\right)\end{aligned}\Rightarrow\overrightarrow{u}\times\overrightarrow{v}=\left(1\times0,2\times0,0\times\left(-2\right)\right)=\left(0,0,0\right)\\\text{Man kan i normala fall inte multiplecera vektorer!}\end{align}$$
+		-