Föreläsning 30 september 2012

Skalärmultiplikation

En skalärmultiplikation

{a,b,c}.{1,2,3}

Definition och figurer (R^2)

vektor1={3,-1};
vektor2={2,6};

Skalärmultiplikation

vektor1.vektor2

Vektorpilar

?Arrow

pil1=Graphics[Arrow[{{0,0},vektor1}]]
pil2=Graphics[Arrow[{{0,0},vektor2}]];

Figur

Show[{pil1,pil2},Axes->True]

Definition och figurer (R^3)

vektor1R3={3,-4,11};
vektor2R3={1,-2,-1};

Skalärmultiplikation

vektor1R3.vektor2R3

Pilar

pil1R3=Graphics3D[Arrow[{{0,0,0},vektor1R3}]]
pil2R3=Graphics3D[Arrow[{{0,0,0},vektor2R3}]];

Figurer

Show[{pil1R3,pil2R3},Axes->True,AspectRatio->Automatic]

Show[{pil1R3,pil2R3},Axes->True,AxesOrigin->{0,0,0}, Boxed->False, PlotRange->{{-12,12},{-12,12},{-12,12}},AxesLabel->{{x},{y},{z}},AspectRatio->Automatic]

Vektorn t v

Manipulate[
Show[Graphics[Arrow[{{0,0},t{x,y}}]],Axes->True,PlotRange->{{-25,25},{-25,25}}],
{{x,3},-9,9,1},
{{y,5},-9,9,1},{{t,1},-9,9}]

Vektorn p+ t v

Manipulate[
Show[Graphics[Arrow[{{p1,p2},{p1+t x,p2+t y}}]],Axes->True,PlotRange->{{-25,25},{-25,25}},AspectRatio->Automatic],
{{x,3},-9,9,1},
{{y,5},-9,9,1},{{t,4},-9,9},
{{p1,3},-9,9,1},
{{p2,3},-9,9,1}]

Längd och norm

En vektor

v1={-2,4,1}

Normen av vektorn

Norm[v1]

En vektor till

Normerad vektor

Vektorn:

v2={13/7,6.3,11}

Normen

normenv2=Norm[v2]

Normeringen

v2Normerad=v2/normenv2

Test, vad är normen för den normerade vektorn

Norm[v2Normerad]

mm

Plan

plan1=ContourPlot3D[x+2y-3z==0,{x,-10,10},{y,-10,10},{z,-10,10}]

plan2=ContourPlot3D[{1,0,0}.{x,y,z}==0,{x,-10,10},{y,-10,10},{z,-10,10}]

Show[plan1,plan2]

Skärning mellan plan

Clear[x,y,z]

Reduce[{x+2y-3z==0,x==0}]

Solve[{x+2y-3z==0,x==0}]