Chapter 6 - Basic Vector Algebra Topics

Section 6.3 - Cross Product

Example 6.3.1

U={-3,7,11};V={6,-1,9};

Det[{{"I","J","K"},U,V}]

Cross[U,V]

Cross[V,U]

Cross[U,V].U

Cross[U,V].V

W={18,-10,-15};

Det[{W,U,V}]

W.Cross[U,V]

ArrowPlots=Graphics3D[{Arrowheads[.07],Thickness[.012],Black,Arrow[{{0,0,0},U}],Blue,Arrow[{{0,0,0},V}],Red,Arrow[{{0,0,0},0.2 Cross[U,V]}]}];

TxtPlots=Graphics3D[{Black,Text["U",{-3,4,9}],Text["V",{6,-1,10}],Text["U x V",{10,10,-5}]}];

RightAnglePlots=Graphics3D[{Black,Polygon[{{0,0,0},{-3/4,7/4,11/4},{-3/4+74/40,7/4+93/40,11/4-39/40},{74/40,93/40,-39/40}}],Black,Polygon[{{0,0,0},{6/4,-1/4,9/4},{6/4+74/40,-1/4+93/40,9/4-39/40},{74/40,93/40,-39/40}}]}];

Show[ArrowPlots,TxtPlots,RightAnglePlots,Axes-> True]

Commands after Example 6.3.1

U={(Subscript[u, 1]),(Subscript[u, 2]),(Subscript[u, 3])};V={(Subscript[v, 1]),(Subscript[v, 2]),(Subscript[v, 3])};

Expand[((((Subscript[u, 2]) (Subscript[v, 3])−(Subscript[v, 2]) (Subscript[u, 3])))^2)+((((Subscript[u, 1]) (Subscript[v, 3])−(Subscript[v, 1]) (Subscript[u, 3])))^2)+((((Subscript[u, 1]) (Subscript[v, 2])−(Subscript[v, 1]) (Subscript[u, 2])))^2)]

NormU=U.U

NormV=V.V

RealDotUV=U.V

Expand[NormU NormV-(RealDotUV^2)]

Example 6.3.2

V={9,-5,-13};W={-4,7,2};

VxW=Cross[V,W]

Area1=1/2 Norm[VxW]

Area2=1/2 Sqrt[((Norm[V])^2) ((Norm[W])^2)-(((V.W))^2)]

θ=N[ArcCos[(V.W)/(Norm[V] Norm[W])]]

N[Norm[VxW]]

LenghVxW=Norm[V] Norm[W] Sin[θ]

Mathematica Problem 1

Manipulate[Column[{Show[Graphics3D[{Blue,Thickness[.01],Arrow[{{0,0,0},{1,0,0}}],Darker[Green],Arrow[{{0,0,0},{Cos[t],Sin[t],0}}],Red,Arrow[{{0,0,0},Cross[{1,0,0},{Cos[t],Sin[t],0}]}]}],PlotRange->{{-1,1},{-1,1},{-1,1}},BoxRatios-> {1,1,1},Axes->  True,AxesEdge-> {Automatic,{1,-1},{-1,-1}},AxesLabel->{"x","y","z"},Ticks-> {{-1,0,1},{-1,0,1},{-1,0,1}}],Row[{"The blue vector {1,0,0} crossed with the green vector" ,Chop[N[{Cos[t],Sin[t],0}]],"is the red vector" Chop[N[Cross[{1,0,0},{Cos[t],Sin[t],0}]]]}],"which has a length of" Chop[N[Norm[Cross[{1,0,0},{Cos[t],Sin[t],0}]]]]}],{{t,0,"Angle "},0,2π,N[π/48]},SaveDefinitions->True]