Lösungen

1

a

Plot[ArcCos[x],{x,-1,1}]

Pi /Degree//N

pO={0,0,0};p1={3,2,1};p2={-3,2,1};p3={3,-2,1};p4={3,2,-1};
winkelR[p1_,p2_]:= ArcCos[p1.p2/Norm[p1]/Norm[p2]];
winkelN[p1_,p2_]:= ArcCos[p1.p2/Norm[p1]/Norm[p2]]//N;
winkelD[p1_,p2_]:= ArcCos[p1.p2/Norm[p1]/Norm[p2]]/Degree//N;
winkel[p1_,p2_]:= {winkelR[p1, p2],winkelN[p1, p2] " Rad",winkelD[p1, p2] " Grad", 
(180-winkelD[p1, p2]) " Grad"}

winkel[p1,p2]

winkel[p1,p3]

winkel[p1,p4]

winkel[p2,p3]

winkel[p2,p4]

winkel[p3,p4]

b

inhaltE[p1_,p2_]:=Norm[Cross[2p1,2 p2]]/2;
inhaltN[p1_,p2_]:=inhaltE[p1,p2]//N;
inhalt[p1_,p2_]:={inhaltE[p1,p2] ," = ", inhaltN[p1,p2], inhaltN[p1,p2]/4};

inhalt[p1,p2]

inhalt[p1,p3]

inhalt[p1,p4]

inhalt[p2,p3]

inhalt[p2,p4]

inhalt[p3,p4]

2

v={1,2,-1}; b1={2,3,-1}; b2={2,3,1}; b3={2,-1,2};
Solve[v==λ b1+μ b2+ν b3,{λ ,μ,ν}]

N[%]

3

Det[{b1,b2,b3}]

4

r1={1,2,-1}; r2={2,2,1}; a1={3,1,1}; a2={1,-1,4};
kuerzesterAbstand=Abs[Det[{r2-r1,a1,a2}]/Norm[Cross[a1,a2]]]

%//N

5

a

vec[x_,y_,z_]:={x,y,z};
{vec[x,y,z].a1==0,vec[x,y,z].a2==0,vec[x,y,z].r2==10}

solv=Solve[{vec[x,y,z].a1==0,vec[x,y,z].a2==0,vec[x,y,z].r2==10},{x,y,z}] // Flatten

%//N

b

vec[x_,y_,z_]:={x,y,z} /. solv;  vec[x,y,z]

Norm[Cross[vec[x,y,z],r2]]

%//N

c

winkel[vec[x,y,z],r2]

6

a

w={{2},{-1},{2}}; w//MatrixForm

dreh[ϕ_]:={{Cos[ϕ],-Sin[ϕ],0},{Sin[ϕ],Cos[ϕ],0},{0,0,1}}; dreh[ϕ]//MatrixForm

dreh[Pi/2].{{1},{0},{0}}//MatrixForm

Table[dreh[n 2Pi/5].w//MatrixForm,{n,1,5}] //Simplify

Table[dreh[n 2Pi/5].w//MatrixForm,{n,1,5}] //Simplify// ExpandAll

N[%]

b

(dreh[2Pi/5].w)

inhalt=5  Transpose[(dreh[2Pi/5].w)].(dreh[2*2Pi/5].w)/2 //Simplify

N[%]