Lösungen

Remove["Global`*"]

Uebung 1

rg1[t_] := {-2, 4} + t {7, 2};
rg2[t_] := {8, 3} + t {-4, 5};

s1a = 
      Solve[rg1[λ] == rg2[μ], {λ, μ}] // 
        Flatten

s1a // N

res1a = {rg1[λ], rg2[μ]} /. s1a

res1a // N

ParametricPlot[Evaluate[rg1[t]], {t, -4, 4}];

ParametricPlot[
        Evaluate[{rg1[t], rg2[t]}], {t, -4, 4}];

h[t_] := 
      rg1[t] + {-2, 7}/Sqrt[(-2)^2  +  7^2] 4

h[t] // N

ParametricPlot[
        Evaluate[{rg1[t], rg2[t], h[t]}], {t, -4, 4}];

s1b = 
      Solve[h[λ] == rg2[μ], {λ, μ}] // 
        Flatten

s1b // N

{h[λ], rg2[μ]} /. s1b  // N

Uebung 2

a

g[t_] := {-2, 4, 1} +  t {7, 2, -1}

Φ[λ_, μ_] := {1, 
          0, -2} + λ {4, 1, 4} + μ {-3, -2, 
            1}

s2a = 
      Solve[g[t] == Φ[λ, μ], {t, λ, μ}] // 
        Flatten

c

res2a // N

b

Ψ[λ_, μ_] := {0, 1, 
          1} + λ {-1, 2, 3} + μ {1, -1, 1}

s2b = 
      Solve[Ψ[λ, μ] == Φ[ζ, η], {μ, ζ, η}] // Flatten

res2b = {Ψ[λ, μ], Φ[ζ, η]} /. s2b

res2b // N

gs[λ_] := res2b[[1]];  gs[λ]

ParametricPlot3D[Evaluate[gs[λ]], {λ, -1.8, 0}, 
        ViewPoint -> {2.265,  1.593,  1.945}];

Solve[gs[λ] == {x, y, 0}, {x, y, λ}]  // 
      Flatten

% // N

Solve[gs[λ] == {x, 0, z}, {x, z, λ}]  // 
      Flatten

% // N

Solve[gs[λ] == {0, y, z}, {y, z, λ}]  // 
      Flatten

% // N

Uebung 3

a

r = Sqrt[1^2 + 2^2 + 3^2]

α = ArcCos[1/r]

N[α]

β = ArcCos[2/r]

N[β]

γ = ArcCos[3/r]

N[γ]

b

x = 1; y = 2; z = 3;

aStrich = {x  Cos[φ] - y  Sin[φ], 
        x  Sin[φ] + y  Cos[φ], z}

φ = π/7;  aStrich

% // N

Uebung 4

a

Remove[x, y, z]

Ψ[λ_, μ_] := {0, 1, 
          1} + λ {-1, 2, 3} + μ {1, -1, 1}

s4 = 
      Solve[Ψ[λ, μ] == {x, y, 
              z}, {λ, μ, x}]  // Flatten

x1 = x /. s4

equat = (x - x1 == 0 // Expand)

(equat[[1]]*5) // Expand

Remove[h]

h[{x_, y_, 
          z_}] := (-3 + 5  x + 4  y - z) *1/
            Sqrt[(5)^2 + (4)^2 + (-1)^2]  // 
        Simplify

?h

h[{x, y, z}]

h[{0, 1, 1}]

h[Ψ[λ, μ]] // Simplify

d = h[{15, 20, 18}]

d // N

b

gH[t_] := {15, 20, 18} +  t {5, 4, -1}

s4b = 
      Solve[gH[t] == Ψ[λ, μ], {t, λ, μ}] // Flatten

res4b = {gH[t], Ψ[λ, μ]} /. s4b

res4b // N

Remove["Global`*"]

Uebung 5

p = (7 - 4) (7 + 4)

solv1 = Solve[t^2 == 33, {t}] // Flatten

t1 = t /. solv1[[2]]

N[t1]

y = Sqrt[4^2 - 2^2]

a = Sqrt[y^2 + (2 + 7)^2]

solv2 = Solve[p == a b, {b}] // Flatten

s = a - b /.  solv2

N[%]

Uebung 6

Remove[x, y]

Solve[{3  x + 4  y == 7, 2  x - 3  y == 5}, {x, 
        y}]

D0 = Det[{{3, 4}, {2, -3}}]

D1 = Det[{{4, -7}, {-3, -5}}]

x = D1/D0

D1 = Det[{{-7, 3}, {-5, 2}}]

y = D1/D0

Uebung 7

a = {a1, a2, a3}; b = {b1, b2, b3}; Cross[a, b]

a = {1, 2, 3}; b = {4, -1, 5}; n = Cross[a, b]

Len[n_] := Sqrt[n . n]

A = Len[n]

A // N

c = {-4, -3, -2};

h = c . n/Len[n]

v = A h

Det[{a, b, c}]

-> V = Det[.....]  !!!!!!

Uebung 8

Remove[x, y, z, a, b, n, xD, h]

a = {1, 2, 3}; b = {4, -1, 5}; 
    n = Cross[a, b]/Norm[Cross[a, b]]

%//N

n . {x, y, z} + dD == 0

solv = 
      Solve[{n . {x, y, z} + dD == 0, x == -4, 
            y == -3, z == -2}, {dD}] // 
        Flatten

%//N

dOrigo = dD /. solv

h[{x_, y_, z_}] := n . {x, y, z} + dOrigo

h[{17, 18, 19}]

%//N

Uebung 9

Remove[g, t, Θ, λ, μ]

g[t_] := {-2, 4, 1} + t {7, 2, 2};
Θ[λ_, μ_] := {8, 
            3, -1} +  λ {-4, 5, 1} + μ {7, 2, 
              2};

Solve[{g[
            t] == Θ[λ, μ]}, {λ, μ}]

V = a * h  ->   h = V / A,   V = Det[....],  A = |Cross[....]|

d = 
      Det[{{-4, 5, 1}, {7, 2, 2},  {8, 3, -1} - {2, 4, 1}}]/
        Len[ Cross[{-4, 5, 1}, {7, 2, 2}]]

N[%]