Lösungen / Solutions

1

pA = {8, 9, 10}; n = {2, -3, 4}; 
    v[{x_, y_, z_}] = {x, y, 
        z};   hNF[{x_, y_, z_}] := (v[{x, y, z}] . n + 5)/
        Sqrt[n . n];
{pA, n, v[{x, y, z}], hNF[{x, y, z}]}

a

aAL = hNF[pA]

% // N

b

g[t_] := pA + t n;  g[t]

solvpL = Solve[hNF[g[t]] == 0, {t}] // Flatten

pL =  g[t] /. solvpL

% // N

c

vLen[v_] := Sqrt[v . v]

aL = vLen[pL]

% // N

aA = vLen[pA]

% // N

2

a  Die Drehung wird hier der Einfachheit halber mit Matrizen gerechnet.

m = {{a, b}, {c, d}}; u = {u1, u2};  m . u  // MatrixForm

pP = {3, 4}; alpha = Pi/6;  
    dreh[alp_] := {{Cos[alp], -Sin[alp]}, {Sin[alp], 
          Cos[alp]}}
pP1 = dreh[alpha] . pP

pP1 // N

pP2 = {pP1[[1]], -pP1[[2]]}

pP2 // N

b

gpP2[t_] := pP2 t;  gpP2[t]

solvt = Solve[ gpP2[t][[1]] == pP[[1]], {t}] // 
          Simplify // Flatten

N[%]

gpP2[t] /. solvt  // Simplify

N[%]

3

pA = {2, 1}; pB = {9, 5}; pC = {4, 8}; vAB = pB - pA; vBC = pC - pB; 
    vCA = pA - pC; pC1 = pA +  1/3 vAB;  pA1 = pB + 1/4 vBC;  
    pAll = {pA, pB, pC, pC1, pA1}

N[%]

a

list1 = Nest[List, pAll, 1] // Transpose

?Apply

list2 = Apply[Point, list1, 1]

list3 = Prepend[list2, PointSize[0.03]]

Show[Graphics[list3]];

b

vAC1B = -(1/3)/(2/3);  vBA1C = -(1/4)/(3/4);  
    vCB1A = -1/(vAC1B *  vBA1C)

pB1 = pC + 6/7 vCA

N[%]

list4 = Append[list3, Point[pB1]]

lin = {Line[{pA, pB}], Line[{pA, pC}], Line[{pC, pB}], Line[{pA, pA1}], 
        Line[{pB1, pB}], Line[{pC, pC1}]}

list5 = Join[lin, list4]

Show[Graphics[list5]];

Das gesuchte Verhältnis ist  1:6

4

a

m = {3, 2}; 
    r = 4;   k[x_, 
        y_] := ((m - {x, y}) . (m - {x, y}) == 
          r^2);   k[{x_, y_}] :=  k[x, y]; 
    g[lamb_] := {1, 1} + lamb {3, 2};
elemente = {Circle[m, r], PointSize[0.03], Point[m], 
          Line[{g[-1], g[2]}]};
Show[Graphics[elemente], 
        AspectRatio -> Automatic];

b

k[g[lamb]]

solv4 = Solve[k[g[lamb]], {lamb}] // Flatten

N[%]

{tP1, tP2} = {g[lamb] /. solv4[[1]], 
        g[lamb] /. solv4[[2]]}

N[%]

t1[x_, 
          y_] := ((m - tP1) . (m - {x, y}) - r^2 == 
            0);
 t2[x_, 
          y_] := ((m - tP2) . (m - {x, y}) - r^2 == 
            0);

solv41 = Solve[t1[x, y], {y}] // Simplify // Flatten

solv42 = Solve[t2[x, y], {y}] // Simplify // Flatten

g1[x_] := y /. solv41;  g2[x_] := y /. solv42;

{g1[x], g2[x]} // N

plot1 = 
        Plot[{g1[x], g2[x]}, {x, -1, 8}, Epilog -> elemente, 
          AspectRatio -> Automatic];

solv43 = Solve[g1[x] == g2[x], {x}] // Simplify // 
        Flatten

x1 = x /. solv43

y1 = g1[x] /. solv43  // Simplify

y2 = g2[x] /. solv43  // Simplify

pQ = {x1, y1}

c

q[x_, 
          y_] := ((m - pQ) . (m - {x, y}) - r^2 == 
            0);

solv44 = Solve[q[x, y], {y}] // Simplify // Flatten

g3[x_] := y /. solv44;  g3[x]

plot2 = 
        Plot[{g1[x], g2[x], g3[x]}, {x, -1, 8}, Epilog -> elemente, 
          AspectRatio -> Automatic];

plot3 = 
        Plot[{g3[x]}, {x, -1, 8}, Epilog -> elemente, 
          AspectRatio -> Automatic];