Lösungen / Solutions

1

Remove[p]

m = {0, 0}; 
    q[s_] = {Cos[s Pi/2 /3], 
        Sin[s Pi/2 / 3]};
vLen[v_] := Sqrt[v . v]; p[k_] := q[k]/vLen[q[k]]

r = 2;
s1 = 
        Show[Graphics[{Circle[{0, 0}, 1], Circle[{0, 0}, r], 
              Line[{p[0], -p[0]}], Line[{r p[1], -r p[1]}], 
              Line[{p[2], -p[2]}], Line[{r p[3], -r p[3]}], 
              Line[{p[4], -p[4]}], Line[{r p[5], -r  p[5]}], 
              Thickness[ .02], 
              Line[{p[0], r p[1], p[2], r p[3], p[4], r p[5], p[6], 
                  r p[7], p[8], r p[9], p[10], r p[11], p[12]}]}], 
          AspectRatio -> Automatic];

r = 3;
s1 = 
        Show[Graphics[{Circle[{0, 0}, 1], Circle[{0, 0}, r], 
              Line[{p[0], -p[0]}], Line[{r p[1], -r p[1]}], 
              Line[{p[2], -p[2]}], Line[{r p[3], -r p[3]}], 
              Line[{p[4], -p[4]}], Line[{r p[5], -r  p[5]}], 
              Thickness[ .02], 
              Line[{p[0], r p[1], p[2], r p[3], p[4], r p[5], p[6], 
                  r p[7], p[8], r p[9], p[10], r p[11], p[12]}]}], 
          AspectRatio -> Automatic];

s2 = 
        Show[Graphics[{Line[{p[0], 2  p[1], p[2], 2  p[3], p[4], 2  p[5], 
                  p[6], 2  p[7], p[8], 2  p[9], p[10], 2  p[11], p[12]}]}], 
          AspectRatio -> Automatic];

Remove[r]

q[s_, r_] = r {Cos[s Pi/2 /3], Sin[s Pi/2 / 3]}

q[1, 2] - q[1, 1]

Det[{q[0, 1], q[1, 1]}]

Det[{q[1, 2] - q[1, 1], q[1, 2] - q[0, 1]}]

6*(Det[{q[0, 1], q[1, 1]}] + 
          Det[{q[1, 2] - q[1, 1], q[1, 2] - q[0, 1]}])

r2

6*(Det[{q[0, r1], q[1, r1]}] + 
          Det[{q[1, r2] - q[1, r1], q[1, r2] - q[0, r1]}])

Det[{q[1, r2] - q[1, r1], q[1, r2] - q[0, r1]}]

Det[{q[0, r1], q[1, r1]}]

2

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

h[{x_, y_, z_}] := (4  x - 2  y + 5  z - 2)/
        vLen[{4, -2, 5}]

m = {9, 12, 15}

h[m]

h[m] // N

h[m] - 2 // N

g[t_] := m + t {4, -2, 5};  g[t]

h[g[t]] // Simplify

solv = Solve[h[g[t]] == 0, {t}] // Flatten

N[%]

(h[g[t]] /. solv) // Simplify

(g[t] /. solv) // Simplify

N[%]

3

gl[x_, y_, α_, β_] := {4  x - α y == 1, 
        8  x + β y == 1}

Solve[gl[x, y, α, β], {x, y}]

Solve[gl[x, y, 1, 2], {x, y}]

N[%]

g1[ α_, β_] := (2 α + β == 
          0)

Solve[g1[1, β], {β}]

Solve[g1[α, β], {α, β}]

4

a

gl1[x0_, y0_, 
        r_] := ((0 - x0)^2 + (6 - y0)^2 == r^2)

gl2[x0_, y0_, 
        r_] := ((-4 - x0)^2 + (0 - y0)^2 == 
          r^2)

gl3[x0_, y0_, 
        r_] := (({-8, 0} - {0, 6}) . ({0, 6} - {x0, 
                  y0}) == 0)

solv = 
      Solve[{gl1[x0, y0, r], gl2[x0, y0, r], gl3[x0, y0, r]}, {x0, y0, 
          r}]

solv // N

Remove["Global`@*"]

b

p0 = {-8, 0} ; pT = {0, 6}; 
    pQadr = (-8)^2 + 6^2

d1 = 4; d2 = pQadr/d1

p1 = {-4, 0}

p0 + {d2, 0}

p2 = p1 + {d2 - d1, 0}

senkr[v_] := {-v[[2]], v[[1]]}

h1[t1_] := pT + t1  senkr[(pT - p0)]; h1[t1]

h2[t2_] := (p1 + p2)/2 + t2 {0, 1}; h2[t2]

solv2 = Solve[h1[t1] == h2[t2], {t1, t2}] // Flatten

h1[t1] /. solv2

h2[t2] /. solv2

N[%]