M3.V.12 A2 AB Geraden im Raum

Geraden mit Vektoren beschreiben
Sie kennen aus der Mittelstufe die eindeutige Berechnung einer Geradengleichung aus zwei Punkten [math]\overrightarrow{P_1}=\begin{pmatrix} x_1 \\ y_1 \end{pmatrix}[/math] und [math]\overrightarrow{P_2}=\begin{pmatrix} x_2 \\ y_2 \end{pmatrix} [/math], die auf der Geraden liegen.[br][img]data:image/png;base64,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[/img][br]Alle Punkte [math]\overrightarrow{P}=\begin{pmatrix} x \\ y \end{pmatrix}[/math], die auf der Geraden liegen, erfüllen die Geradengleichung [math]y=\frac{y_2-y_1}{x_2-x_1} \cdot (x-x_1)+y_1[/math].[br]Man erkennt in der Gleichung die [b]Änderung [/b]der Koordinaten zwischen den beiden Punkten [math](y_2-y_1)[/math] und [math](x_2-x_1)[/math] und die Koordinaten von [math]P_1[/math] als [b]Bezugspunkt[/b].[br]Ganz ähnlich kann man Geraden auch mit Vektoren und deren geometrischen Deutungen beschreiben - in 2D und 3D: man benötigt einen Bezugspunkt und einen Pfeil, um die Änderungen zu beschreiben.
Aufgabe 1
Erkunden Sie das nachfolgende Applet und beschreiben Sie, wie man mithilfe von Punkten und Vektoren eine Geradengleichung aufstellt. Erläutern Sie die einzelnen Bestandteile der Gleichung.
[b][size=150][color=#cc0000]||[/color][color=#1155cc] Benutzerhinweise zum obigen Applet[/color][/size][/b][br][b][size=150][color=#cc0000]|| [/color][/size][/b]Ein Häkchen bei [color=#00ff00]mehr Repräsentanten[/color] blendet Pfeildarstellungen desselben Vektors ein. [br][b][size=150][color=#cc0000]|| [/color][/size][/b]Ein Häkchen bei [color=#cc0000]Punkt [/color]und [color=#0000ff]Verschiebungsvektor [/color]zeichnet einen Punkt und von diesem aus eine [br][b][size=150][color=#cc0000]|| [/color][/size][/b]Pfeildarstellung des 1.5-fachen des Vektors [math]\vec{v}[/math] ein. Das Pfeilende markiert den Punkt [color=#0000ff][math]\vec X [/math][/color]. [br][b][size=150][color=#cc0000]|| [/color][/size][/b]Der Faktor [color=#0000ff]t=1.5[/color] lässt sich per Schieberegler ändern.[br][b][size=150][color=#cc0000]|| [/color][/size][/b]Mit den Schaltern [color=#6aa84f]an [/color][color=#cc0000]aus [/color]unter Spur [math]\vec X [/math] hinterlässt der Punkt [color=#0000ff][math]\vec{X}[/math][/color] (Schieberegler) eine Spur.[br][b][size=150][color=#cc0000]|| [/color][/size][/b]Ein Häkchen bei Gerade zeichnet eine Gerade und bei Geradengleichung wird diese dazu angezeigt.[br][b][size=150][color=#cc0000][b][size=150][color=#cc0000][br]||[/color][/size][/b] [/color][/size][/b]Wenn man oben rechts im Applet auf [img]https://juergen-roth.de/images/icons/jr/Schaltflaeche_neu_laden.png[/img] klickt, wird das Applet auf seinen Ausgangszustand zurückgesetzt. [br][b][size=150][color=#cc0000][b][size=150][color=#cc0000]||[/color][/size][/b] [/color][/size][/b]Wenn man unten rechts im Applet auf [img]https://juergen-roth.de/images/icons/jr/ggb_vollbild_icon.png[/img] klickt, wird das Applet im Vollbild dargestellt.[br][br]
Aufgabe 2
Erkunden Sie das nachfolgende Applet und geben Sie in der Eingabezeile die Geradengleichung in der Form [math] h(t)=A+t\cdot v [/math] ein. Erzeugen Sie dann auf möglichst vielen Wegen eine Gerade durch die Punkte [math]C=(4,4,1)[/math] und [math]D=(0,-1,2)[/math].[br]Notieren Sie schließlich drei verschiedene Vorgehen, um in GeoGebra 3D eine Gerade zu erzeugen.
In GeoGebra 3D
[b][size=150][color=#cc0000]||[/color][color=#1155cc] Benutzerhinweise zum obigen Applet[/color][/size][/b][br][b][size=150][color=#cc0000]|| [/color][/size][/b]Vektoren und Punkte werden in GeoGebra beide als Liste von Einträgen [code]...=(1,2,3)[/code] eingegeben.[br][b][size=150][color=#cc0000]|| [/color][/size][/b]Bei Großbuchstaben interpretiert GeoGebra dies als Punkt, bei Kleinbuchstaben als Pfeil.[br][b][size=150][color=#cc0000]|| [/color][/size][/b]Ein weiterer Punkt lässt sich dynamisch als Summe aus Punkt und Vielfachem eines Vektors erzeugen. [br][b][size=150][color=#cc0000]|| [/color][/size][/b]Ist der Bezeichner des Vielfachen noch nicht vergeben - wie hier k - interpretiert GeoGebra diesen[br][b][size=150][color=#cc0000]|| [/color][/size][/b]als Parameter und erzeugt automatisch einen Schieberegler.[b][size=150][color=#cc0000][b][size=150][color=#cc0000][br]||[/color][/size][/b] [/color][/size][/b]Wenn man oben rechts im Applet auf [img]https://juergen-roth.de/images/icons/jr/Schaltflaeche_neu_laden.png[/img] klickt, wird das Applet auf seinen Ausgangszustand zurückgesetzt. [br][b][size=150][color=#cc0000][b][size=150][color=#cc0000]||[/color][/size][/b] [/color][/size][/b]Wenn man unten rechts im Applet auf [img]https://juergen-roth.de/images/icons/jr/ggb_vollbild_icon.png[/img] klickt, wird das Applet im Vollbild dargestellt.[br][br]
[i][u]Quellen: [/u][br]Susanne Digel, Jürgen Roth.[/i]
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Information: M3.V.12 A2 AB Geraden im Raum