Lösungen / Solutions

1

a

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

V1 > 0  !!!

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

V1 + V2

solva = Solve[V1 + V2 == 20, {α}] // Flatten

V2 /. solva

O.k.

b

V1 < 0  !!!

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

V1 + V2

solvb = Solve[V1 + V2 == 20, {α}] // Flatten

V2 /. solva

O.k.,     wie a) / comme a)

2

pn[2] = p[2] + p[3]

p[2  a] = {-3, 3, 8};
p[2  b] = {-3, -3, 8};
p[2  c] = {-3, -3, -8};
V1 = 
      1/6 (Det[{(p[2  a] - pn[2]), (p[2  b] - pn[2]), (p[2  c] - 
                  pn[2])}])

3

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

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

\(\(\(===\)\(\(>\)\(\ \)\(α\)\(\ \ \)\(≠\)\(\ \)\(\(-\ \(1\/4\)\)\ \ \ \((ist\ bei\ 1. \ erfüllt)\)\)\)\)\)

solv = Solve[-((-4 β + 3 γ - 
                      2 δ)/(1 + 
                      4 α)) == -((3 β - 
                      2 γ + α γ + δ - 
                      2 α δ)/(1 + 
                      4 α)), { β}] // Simplify // 
        Flatten

solv /. α -> -44  // Simplify

solv /. α -> 54  // Simplify

4

Komplex schreiben in 2 Koordinaten

pC[1] = p[3][[1]] + I p[3][[2]]

Drehung um -20 Grad

qC =  pC[1]* E^(I  (-20)/360  *2  Pi)

qC // N

Wieder 3-dimensional schreiben

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

q // N

Wieder komplex schreiben in 2 neuen (anderen) Koordinaten

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

qCC // N

Drehung um 40 Grad

sC =  qCC* E^(I  40/360  *2  Pi)

sC // N

Wieder 3-dimensional schreiben

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

s // N

Distanz rechnen

dVec = p[3] - s

dVec // N

d = Sqrt[dVec . dVec]

d // N