Lösungen / Solutions

1

p[1] = {1, 1, 1};
p[2] = {1, 2, 3};
p[3] = {2, 1, 3};
p[4] = {1, 3, 2};
p[5] = {4, 4, 4};
V1 = 
      1/6 (Det[{(p[2] - p[1]), (p[3] - p[1]), (p[4] - 
                  p[1])}])

V2 = -1/
          6 (Det[{(p[2] - p[5]), (p[3] - p[5]), (p[4] - 
                  p[5])}])

V1 + V2

2

p[2] = {1, 2, 3};
p[2  a] = {-2, 1, 3};
p[2  b] = {-2, -1, 3};
p[2  c] = {-2, -1, -3};
V1 = 
      1/6 (Det[{(p[2  a] - p[2]), (p[2  b] - p[2]), (p[2  c] - 
                  p[2])}])

3

vx = {x, y, z};
gl = {p[1] . vx == α, p[2] . vx == β, 
        p[3] . vx == γ}

Solve[gl, {x, y, z}]

Solve[
      (1/3) (3 α - 2 β + γ) == 
        (1/3) (-3 α + β + γ), { α}]

4

Komplex schreiben in 2 Koordinaten

pC[1] = p[1][[1]] + I p[1][[2]]

Drehung um 15 Grad

qC =  pC[1]* E^(I  15/360  *2  Pi)

qC // N

Wieder 3-dimensional schreiben

q = {Re[qC], Im[qC], p[1][[3]]}

q // N

Wieder komplex schreiben in 2 neuen (anderen) Koordinaten

qCC = q[[3]] + I q[[1]]

qCC // N

Drehung um -22 Grad

sC =  qCC* E^(-I  22/360  *2  Pi)

sC // N

Wieder 3-dimensional schreiben

s = {Im[sC], q[[2]], Re[sC]}

s // N

Distanz rechnen

dVec = p[1] - s

dVec // N

d = Sqrt[dVec . dVec]

d // N