FHTW Typst bei Abbildung 4
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48
FHTW.typ
48
FHTW.typ
@@ -38,36 +38,32 @@ Also: $RR={arrow(a)=(a_1,a_2) | a_1 in RR, a_2 in RR }$
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=== Addition von Vektoren in der Ebene
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- liefert wieder einen Vektor
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#table(
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columns: 2,
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stroke: none,
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[- liefert wieder einen Vektor
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- rechnerisch:
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- $arrow(a)=(a_1, a_2)$, $arrow(b)=(b_1, b_2)$
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- $arrow(a)+arrow(b)=(a_1+b_1, a_2+b_2)$
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- $arrow(a)+arrow(b)=(a_1+b_1, a_2+b_2)$
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- Rechenregeln:
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- $arrow(a)+arrow(b)=arrow(b)+arrow(a)$
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- $arrow(a)+(arrow(b)+arrow(c)))$; $(arrow(a)+arrow(b))=arrow(c)$],[zeichnerisch:#abb3()]
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)
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#abb3()
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#pagebreak()
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=== Nullvektor in der Ebene
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- $arrow(0)=(0,0)$
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- Rechenregel: $arrow(a)+arrow(0)=arrow(a)$
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/*1.1.2 Menge aller Vektoren in der Ebene
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=== Subtraktion von Vektoren in der Ebene
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Die Menge aller Vektoren in der Ebene heißt $\mathbb{R}^2$; dabei ist $\mathbb{R}$ die Menge der reellen Zahlen. Also: $\mathbb{R}=\left\{\vec{a}=\left(a_1, a_2\right) \mid a_1 \in \mathbb{R}, a_2 \in \mathbb{R}\right\}$
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1.1.3 Addition von Vektoren in der Ebene
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#table(
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columns: 2,
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stroke: none,
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[- liefert wieder einen Vektor
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- rechnerisch:
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- $arrow(a)=(a_1,a_2)$, $arrow(b)=(b_1, b_2)$
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- $arrow(a)-arrow(b)= (a_1-b_1, a_2-b_2$) ],[#abb4()]
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)
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Die Addition von Vektoren liefert als Ergebnis wieder einen Vektor.
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rechnerisch:
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$$
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\begin{aligned}
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& \vec{a}=\left(a_1, a_2\right), \vec{b}=\left(b_1, b_2\right) \\
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& \vec{a}+\vec{b}=\left(\left(a_1+b_1, a_2+b_2\right)\right)
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\end{aligned}
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$$
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zeichnerisch:
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Rechenregeln:
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$$
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\begin{aligned}
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& \vec{a}+\vec{b}=\vec{b}+\vec{a} \\
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& \vec{a}+(\vec{b}+\vec{c})=(\vec{a}+\vec{b})+\vec{c}
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\end{aligned}
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$$*/
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@@ -1,4 +1,5 @@
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#import "@preview/cetz:0.3.3" //Grafiken
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#set text(font: "New Computer Modern Sans Math")
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#let abb1() = {
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@@ -17,13 +18,11 @@ cetz.canvas(length: 3cm, {
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content: (padding: 1pt)
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)
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grid((-.25, -.25), (2, 2 ), step: 0.5, stroke: gray + 0.2pt)
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//grid((-.25, -.25), (2, 2 ), step: 0.5, stroke: gray + 0.2pt)
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line((-.25, 0), (1.85, 0), mark: (end: "stealth"))
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line((-.05, 0), (1.85, 0), mark: (end: "stealth", scale: 0.8))
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content((), $ x $, anchor: "west")
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line((0, -.25), (0, 1.85), mark: (end: "stealth"))
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line((0, -.05), (0, 1.85), mark: (end: "stealth", scale: 0.8))
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content((), $ y $, anchor: "south")
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for (x, ct) in ((0, $ 0 $), (0.5, $ 1 $), (1, $ 2 $), (1.5, $ 3 $), (1.25, $a_1$)) {
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@@ -44,9 +43,9 @@ cetz.canvas(length: 3cm, {
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line((0,0), (1.25, 1.25),name: "vec",stroke: (paint: red.darken(30%), thickness: 1.25pt))
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line((1.25,0), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.75pt))
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line((1.25,0), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.5pt))
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line((0,1.25), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.75pt))
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line((0,1.25), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.5pt))
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circle((1.25,1.25), radius: 0.025, fill: green.darken(30%), stroke: (paint: green.darken(30%), thickness: 1.25pt), name: "punkt")
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content("punkt.end", anchor: "south-west", padding: (left: 3mm, bottom: 2mm), text(green.darken(30%))[Punkt $(a_1, a_2)$])})
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@@ -68,13 +67,11 @@ circle((1.25,1.25), radius: 0.025, fill: green.darken(30%), stroke: (paint: gree
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content: (padding: 1pt)
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)
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grid((-.25, -.25), (2, 2 ), step: 0.5, stroke: gray + 0.2pt)
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// grid((-.25, -.25), (2, 2 ), step: 0.5, stroke: gray + 0.2pt)
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line((-.25, 0), (1.85, 0), mark: (end: "stealth"))
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line((-.05, 0), (1.85, 0), mark: (end: "stealth", scale: 0.8))
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content((), $ x $, anchor: "west")
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line((0, -.25), (0, 1.85), mark: (end: "stealth"))
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line((0, -.05), (0, 1.85), mark: (end: "stealth", scale: 0.8))
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content((), $ y $, anchor: "south")
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for (x, ct) in ((0, $ 0 $), (0.5, $ 1 $), (1, $ 2 $), (1.5, $ 3 $), (1.25, $a_1$)) {
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@@ -91,11 +88,11 @@ circle((1.25,1.25), radius: 0.025, fill: green.darken(30%), stroke: (paint: gree
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line((0,0), (1.25, 1.25),name: "vec",stroke: (paint: red.darken(30%), thickness: 1.25pt))
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line((1.25,0), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.75pt))
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line((1.25,0), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.5pt))
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line((0,1.25), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.75pt))
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line((0,1.25), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.5pt))
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line((0,0), (1.25, 1.25),name: "vec",mark: (end: "stealth", fill: green.darken(30%)), stroke: (paint: green.darken(30%), thickness: 1.25pt))
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line((0,0), (1.25, 1.25),name: "vec",mark: (end: "stealth", fill: green.darken(30%),scale: 0.8), stroke: (paint: green.darken(30%), thickness: 1.25pt))
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content("vec.end", anchor: "north-west", padding: (left: 3mm, top: -2mm), text(green.darken(30%))[Vektor $arrow(a)$])
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})
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@@ -103,7 +100,7 @@ line((0,0), (1.25, 1.25),name: "vec",stroke: (paint: red.darken(30%), thickness:
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}
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#let abb3() = {
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cetz.canvas(length: 37mm,
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cetz.canvas(length: 40mm,
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{
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import cetz.draw: *
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@@ -119,34 +116,90 @@ line((0,0), (1.25, 1.25),name: "vec",stroke: (paint: red.darken(30%), thickness:
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content: (padding: 1pt)
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)
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grid((-.25, -.25), (2.5 , 2 ), step: 0.5, stroke: gray + 0.2pt)
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//grid((-.25, -.25), (2.5 , 2 ), step: 0.5, stroke: gray + 0.2pt)
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line((-.25, 0), (2.25, 0), mark: (end: "stealth"))
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line((-.05, 0), (80mm, 0), mark: (end: "stealth",scale: 0.9))
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content((), $ x $, anchor: "west")
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line((0, -.25), (0, 1.85), mark: (end: "stealth"))
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line((0, -.05), (0, 70mm), mark: (end: "stealth",scale: 0.9))
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content((), $ y $, anchor: "south")
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for (x, ct) in ((0.25, $ 1 $), (0.5, $ 2$), (0.75, $ a_1 $),(1, $b_1$), (1.25,$5$), (1.5, $6$),(1.75, $a_1+b_1$)) {
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for (x, ct) in ((10mm, $ 1 $), (20mm, $ b_1$), (30mm, $ 3 $),(40mm, $a_1$), (50mm,$5$), (60mm, $a_1 + b_1$),(70mm, $7$)) {
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line((x, 3pt), (x, -3pt))
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content((), anchor: "north", ct)
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}
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for (y, ct) in ((0.25, $ 1 $), (0.5, $ a_2 $), (.75, $ 3 $), (1., $b_2$),(1.25,$5$), (1.5, $6$)) {
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for (y, ct) in ((10mm, $ a_2 $), (20mm, $ 2 $), (30mm, $ 3 $), (40mm, $b_2$),(1.25,$a_2+b_2$), (1.5, $6$)) {
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line((3pt, y), (-3pt, y))
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content((), anchor: "east", ct)
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}
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line((0,0), (0.75, .5),name: "veca",mark: (end: "stealth", fill: blue.darken(30%)), stroke: (paint: blue.darken(30%), thickness: 0.75pt))
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line((0,0), (40mm, 10mm),name: "veca",mark: (end: "stealth", fill: blue.darken(30%),scale:0.9), stroke: (paint: blue.darken(30%), thickness: 0.75pt))
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content("veca.100%", anchor: "west", padding: (left:1mm, bottom:-5mm),text(blue.darken(30%))[$arrow(a)$])
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content("veca.100%", anchor: "west", padding: (left:1mm, bottom:-5mm),text(blue.darken(30%))[$arrow(a)$])
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line((0,0), (1, 1),name: "vecb",mark: (end: "stealth", fill: red.darken(30%)), stroke: (paint: red.darken(30%), thickness: 0.75pt))
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line((40mm,0), (40mm, 10mm ), stroke: (dash: "dotted",paint: blue.darken(30%), thickness: 0.5pt))
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content("vecb.80%", anchor: "east", padding: (left:0mm, bottom:5mm),text(red.darken(30%))[$arrow(b)$])
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line((0,10mm), (40mm, 10mm), stroke: (dash: "dotted",paint: blue.darken(30%), thickness: 0.5pt))
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line((0,0), (20mm, 40mm),name: "vecb",mark: (end: "stealth", fill: red.darken(30%),scale: 0.9), stroke: (paint: red.darken(30%), thickness: 0.75pt))
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//
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content("vecb.80%", anchor: "east", padding: (left:0mm, bottom:5mm),text(red.darken(30%))[$arrow(b)$])
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//line((0,0), (1.5, 1),name: "vecb",mark: (end: "stealth", fill: red), stroke: (paint: red, thickness: 0.75pt))
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line((20mm,0mm), (20mm, 40mm), stroke: (dash: "dotted",paint: red.darken(30%), thickness: 0.5pt))
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line((0mm,40mm), (20mm, 40mm), stroke: (dash: "dotted",paint: red.darken(30%), thickness: 0.5pt))
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})}
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line((0,0), (60mm, 50mm),name: "vec_ab",mark: (end: "stealth", fill: orange ,scale: 0.9), stroke: (paint: orange, thickness: 0.75pt))
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content("vec_ab.100%", anchor: "east", padding: (left:0mm, bottom:5mm),text(green.darken(60%))[$arrow(a) + arrow(b)$])
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line((20mm,40mm), (60mm, 50mm),name: "veca_dotted",mark: (end: "stealth", scale: 0.9), stroke: (dash: "dotted", paint: blue, thickness: 0.75pt))
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line((40mm,10mm), (60mm, 50mm),name: "vecb_dotted",mark: (end: "stealth", scale: 0.9), stroke: (dash: "dotted", paint: red, thickness: 0.75pt))
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content((50mm, 20mm), [Parallelogramm])
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arc(
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(48mm, 18mm),
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radius: 0.25,
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start: -45deg,
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delta: -90deg,
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mark: (
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end: "stealth",
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))
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line((60mm,0mm), (60mm, 50mm),name: "vecab_dotted", stroke: (dash: "dotted", paint: orange, thickness: 0.75pt))
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line((0mm,50mm), (60mm, 50mm),name: "vecab_dotted",stroke: (dash: "dotted", paint: orange, thickness: 0.75pt))
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})}
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#let abb4() = {
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cetz.canvas(length: 40mm,
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{
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import cetz.draw: *
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set-style(
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mark: (fill: black, scale: 2),
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stroke: (thickness: 0.4pt, cap: "round"),
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angle: (
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radius: 0.3,
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label-radius: .22,
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fill: green.lighten(80%),
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stroke: (paint: green.darken(50%))
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),
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content: (padding: 1pt)
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)
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//grid((-.25, -.25), (2.5 , 2 ), step: 0.5, stroke: gray + 0.2pt)
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line((-30mm, 0), (50mm, 0), mark: (end: "stealth",scale: 0.9))
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content((), $ x $, anchor: "west")
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line((0, -.05), (0, 70mm), mark: (end: "stealth",scale: 0.9))
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content((), $ y $, anchor: "south")
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})
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}
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