miniAufgabe("#exopovray_basics2","#solpovray_basics2",
[["", "\ntorus {1, 0.2\n rotate 180*x\n rotate 135*y\n translate <2, 4, 2>\n}
"], ["", "\ntorus {1, 0.2\n rotate 180*y\n rotate 45*x\n translate <2, 3, 4>\n}
"], ["", "\ntorus {1, 0.2\n rotate -90*x\n rotate 135*y\n translate <2, 4, 4>\n}
"], ["", "\ntorus {1, 0.2\n rotate 90*y\n rotate 45*x\n translate <4, 2, 3>\n}
"], ["", "\ntorus {1, 0.2\n rotate -90*x\n rotate 45*y\n translate <2, 4, 2>\n}
"], ["", "\ntorus {1, 0.2\n rotate 90*y\n rotate 45*z\n translate <2, 3, 3>\n}
"], ["", "\ntorus {1, 0.2\n rotate 90*x\n rotate 135*z\n translate <3, 5, 2>\n}
"], ["", "\ntorus {1, 0.2\n rotate 90*z\n rotate 45*x\n translate <1, 2, 4>\n}
"], ["", "\ntorus {1, 0.2\n rotate 180*z\n rotate 45*y\n translate <2, 2, 3>\n}
"], ["", "\ntorus {1, 0.2\n rotate 90*z\n rotate 135*x\n translate <3, 2, 4>\n}
"], ["", "\ntorus {1, 0.2\n rotate -90*z\n rotate 135*y\n translate <2, 5, 3>\n}
"], ["", "\ntorus {1, 0.2\n rotate 90*y\n rotate 135*x\n translate <1, 3, 4>\n}
"], ["", "\ntorus {1, 0.2\n rotate -90*z\n rotate 45*x\n translate <2, 2, 2>\n}
"], ["", "\ntorus {1, 0.2\n rotate -90*z\n rotate 135*x\n translate <1, 4, 4>\n}
"], ["", "\ntorus {1, 0.2\n rotate 180*x\n rotate 45*y\n translate <3, 2, 2>\n}
"]],
" ", "
");
miniAufgabe("#exopovray_basics3","#solpovray_basics3",
[["$A=(0, 0, -3)$, $B=(1, -1, 1)$, $t=\\frac{1}{2}$", "$$\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA} = \\begin{pmatrix}1\\\\ -1\\\\ 4\\end{pmatrix}$$\n$$\\overrightarrow{OP}(t) = \\overrightarrow{OA} + t \\cdot \\overrightarrow{AB} = \\begin{pmatrix}0\\\\ 0\\\\ -3\\end{pmatrix} + t \\cdot \\begin{pmatrix}1\\\\ -1\\\\ 4\\end{pmatrix}$$\n$$\\overrightarrow{OP}\\left(\\frac{1}{2}\\right) = \\begin{pmatrix}\\frac{1}{2}\\\\ -\\frac{1}{2}\\\\ -1\\end{pmatrix}$$\ntranslate <0, 0, -3> + clock * <1, -1, 4>
\n"], ["$A=(2, -2, 2)$, $B=(1, 1, -4)$, $t=\\frac{1}{3}$", "$$\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA} = \\begin{pmatrix}-1\\\\ 3\\\\ -6\\end{pmatrix}$$\n$$\\overrightarrow{OP}(t) = \\overrightarrow{OA} + t \\cdot \\overrightarrow{AB} = \\begin{pmatrix}2\\\\ -2\\\\ 2\\end{pmatrix} + t \\cdot \\begin{pmatrix}-1\\\\ 3\\\\ -6\\end{pmatrix}$$\n$$\\overrightarrow{OP}\\left(\\frac{1}{3}\\right) = \\begin{pmatrix}\\frac{5}{3}\\\\ -1\\\\ 0\\end{pmatrix}$$\ntranslate <2, -2, 2> + clock * <-1, 3, -6>
\n"], ["$A=(2, -4, 3)$, $B=(-4, -2, 2)$, $t=\\frac{1}{5}$", "$$\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA} = \\begin{pmatrix}-6\\\\ 2\\\\ -1\\end{pmatrix}$$\n$$\\overrightarrow{OP}(t) = \\overrightarrow{OA} + t \\cdot \\overrightarrow{AB} = \\begin{pmatrix}2\\\\ -4\\\\ 3\\end{pmatrix} + t \\cdot \\begin{pmatrix}-6\\\\ 2\\\\ -1\\end{pmatrix}$$\n$$\\overrightarrow{OP}\\left(\\frac{1}{5}\\right) = \\begin{pmatrix}\\frac{4}{5}\\\\ -\\frac{18}{5}\\\\ \\frac{14}{5}\\end{pmatrix}$$\ntranslate <2, -4, 3> + clock * <-6, 2, -1>
\n"], ["$A=(-4, -1, -4)$, $B=(4, 2, 1)$, $t=\\frac{1}{2}$", "$$\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA} = \\begin{pmatrix}8\\\\ 3\\\\ 5\\end{pmatrix}$$\n$$\\overrightarrow{OP}(t) = \\overrightarrow{OA} + t \\cdot \\overrightarrow{AB} = \\begin{pmatrix}-4\\\\ -1\\\\ -4\\end{pmatrix} + t \\cdot \\begin{pmatrix}8\\\\ 3\\\\ 5\\end{pmatrix}$$\n$$\\overrightarrow{OP}\\left(\\frac{1}{2}\\right) = \\begin{pmatrix}0\\\\ \\frac{1}{2}\\\\ -\\frac{3}{2}\\end{pmatrix}$$\ntranslate <-4, -1, -4> + clock * <8, 3, 5>
\n"], ["$A=(0, 0, -2)$, $B=(4, -2, -1)$, $t=\\frac{2}{3}$", "$$\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA} = \\begin{pmatrix}4\\\\ -2\\\\ 1\\end{pmatrix}$$\n$$\\overrightarrow{OP}(t) = \\overrightarrow{OA} + t \\cdot \\overrightarrow{AB} = \\begin{pmatrix}0\\\\ 0\\\\ -2\\end{pmatrix} + t \\cdot \\begin{pmatrix}4\\\\ -2\\\\ 1\\end{pmatrix}$$\n$$\\overrightarrow{OP}\\left(\\frac{2}{3}\\right) = \\begin{pmatrix}\\frac{8}{3}\\\\ -\\frac{4}{3}\\\\ -\\frac{4}{3}\\end{pmatrix}$$\ntranslate <0, 0, -2> + clock * <4, -2, 1>
\n"], ["$A=(-3, -2, -4)$, $B=(-4, -4, -1)$, $t=\\frac{4}{5}$", "$$\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA} = \\begin{pmatrix}-1\\\\ -2\\\\ 3\\end{pmatrix}$$\n$$\\overrightarrow{OP}(t) = \\overrightarrow{OA} + t \\cdot \\overrightarrow{AB} = \\begin{pmatrix}-3\\\\ -2\\\\ -4\\end{pmatrix} + t \\cdot \\begin{pmatrix}-1\\\\ -2\\\\ 3\\end{pmatrix}$$\n$$\\overrightarrow{OP}\\left(\\frac{4}{5}\\right) = \\begin{pmatrix}-\\frac{19}{5}\\\\ -\\frac{18}{5}\\\\ -\\frac{8}{5}\\end{pmatrix}$$\ntranslate <-3, -2, -4> + clock * <-1, -2, 3>
\n"], ["$A=(0, 1, 3)$, $B=(1, 4, -2)$, $t=\\frac{1}{5}$", "$$\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA} = \\begin{pmatrix}1\\\\ 3\\\\ -5\\end{pmatrix}$$\n$$\\overrightarrow{OP}(t) = \\overrightarrow{OA} + t \\cdot \\overrightarrow{AB} = \\begin{pmatrix}0\\\\ 1\\\\ 3\\end{pmatrix} + t \\cdot \\begin{pmatrix}1\\\\ 3\\\\ -5\\end{pmatrix}$$\n$$\\overrightarrow{OP}\\left(\\frac{1}{5}\\right) = \\begin{pmatrix}\\frac{1}{5}\\\\ \\frac{8}{5}\\\\ 2\\end{pmatrix}$$\ntranslate <0, 1, 3> + clock * <1, 3, -5>
\n"], ["$A=(0, -3, -3)$, $B=(1, -4, -2)$, $t=\\frac{1}{5}$", "$$\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA} = \\begin{pmatrix}1\\\\ -1\\\\ 1\\end{pmatrix}$$\n$$\\overrightarrow{OP}(t) = \\overrightarrow{OA} + t \\cdot \\overrightarrow{AB} = \\begin{pmatrix}0\\\\ -3\\\\ -3\\end{pmatrix} + t \\cdot \\begin{pmatrix}1\\\\ -1\\\\ 1\\end{pmatrix}$$\n$$\\overrightarrow{OP}\\left(\\frac{1}{5}\\right) = \\begin{pmatrix}\\frac{1}{5}\\\\ -\\frac{16}{5}\\\\ -\\frac{14}{5}\\end{pmatrix}$$\ntranslate <0, -3, -3> + clock * <1, -1, 1>
\n"], ["$A=(-2, -2, -3)$, $B=(-3, -3, 0)$, $t=\\frac{4}{5}$", "$$\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA} = \\begin{pmatrix}-1\\\\ -1\\\\ 3\\end{pmatrix}$$\n$$\\overrightarrow{OP}(t) = \\overrightarrow{OA} + t \\cdot \\overrightarrow{AB} = \\begin{pmatrix}-2\\\\ -2\\\\ -3\\end{pmatrix} + t \\cdot \\begin{pmatrix}-1\\\\ -1\\\\ 3\\end{pmatrix}$$\n$$\\overrightarrow{OP}\\left(\\frac{4}{5}\\right) = \\begin{pmatrix}-\\frac{14}{5}\\\\ -\\frac{14}{5}\\\\ -\\frac{3}{5}\\end{pmatrix}$$\ntranslate <-2, -2, -3> + clock * <-1, -1, 3>
\n"], ["$A=(3, 0, -2)$, $B=(-4, -1, 0)$, $t=\\frac{1}{3}$", "$$\\overrightarrow{AB} = \\overrightarrow{OB}-\\overrightarrow{OA} = \\begin{pmatrix}-7\\\\ -1\\\\ 2\\end{pmatrix}$$\n$$\\overrightarrow{OP}(t) = \\overrightarrow{OA} + t \\cdot \\overrightarrow{AB} = \\begin{pmatrix}3\\\\ 0\\\\ -2\\end{pmatrix} + t \\cdot \\begin{pmatrix}-7\\\\ -1\\\\ 2\\end{pmatrix}$$\n$$\\overrightarrow{OP}\\left(\\frac{1}{3}\\right) = \\begin{pmatrix}\\frac{2}{3}\\\\ -\\frac{1}{3}\\\\ -\\frac{4}{3}\\end{pmatrix}$$\ntranslate <3, 0, -2> + clock * <-7, -1, 2>
\n"]],
"
", "
");