Chapter 6 - Basic Vector Algebra Topics

Section 6.2 - Dot Product

Example 6.2.1

V={5,-1,2,3};W={-4,0,-5,7};

V.W

Example 6.2.2

V={12,19};W={35,0};

ArrowPlots=Graphics[{Arrowheads[.10],Thickness[.010],Blue,Arrow[{{0,0},V}],Black,Arrow[{{0,0},W}],Red,Arrow[{W,W+(V-W)}]}];

LinePlot=Graphics[{Thickness[.005],Black,Line[{{12,0},V}]}];
TxtPlots=Graphics[{Black,Text["V",{5,10}],Text["W",{16,-1}],Text["V-W",{23,12}],Text["H",{13.5,10}],Text["θ", {4,2}]}];

RightAnglePlot=Graphics[{Black,Rectangle[{12,0},{14,2}]}];

ArcPlot=Graphics[{Black,Thickness[0.010],Circle[{0,0},7,{0,ArcTan[19/12]}]}];

Show[ArcPlot,LinePlot,ArrowPlots,TxtPlots,RightAnglePlot]

Example 6.2.3

P={-8,3,17};Q={10,-6,-9};R={30,13,25};

TrianglePQR=Graphics3D[{Blue,Opacity[.5],Polygon[{P,Q,R}]}];

ArrowPlots=Graphics3D[{Arrowheads[.1],Thickness[.015],Red,Arrow[{P,Q}],Black,Arrow[{P,R}],Blue,Arrow[{R,Q}]}];

TxtPlots=Graphics3D[{Black,Text["PQ",{-3,-2,4}],Text["PR",{11,8,22.5}],Text["RQ",{22,4,8}]}];

Show[TrianglePQR,ArrowPlots,TxtPlots,Axes->True,ViewPoint->{17,-25,20}]

PQ=Q-P

PR=R-P

RQ=Q-R

AngleQ=ArcCos[N[(PQ.PR)/(Norm[PQ] Norm[PR])]]

AngleR=ArcCos[N[(PQ.RQ)/(Norm[PQ] Norm[RQ])]]

AngleP=ArcCos[N[(RQ.-PR)/(Norm[RQ] Norm[PR])]]

AngleQ+AngleR+AngleP

.5 Sqrt[((Norm[PQ])^2) ((Norm[PR])^2)-(((PQ.PR))^2)]

VectorAngle[PQ,PR]

N[VectorAngle[PQ,PR]]

N[VectorAngle[PQ,RQ]]

N[VectorAngle[RQ,-PR]]

Example 6.2.4

V={-4,5,11,-3};U={5,4,3,11};W={11,3,4,5};

{U.V,V.W}

U.W

N[ArcCos[U.W/(Norm[U] Norm[W])]]

Example 6.2.5

u={2-I,1-3 I,I};v={2-5 I,6+7 I,-3};

u.v

v.u

u.Conjugate[v]

v.Conjugate[u]

Norm[u]

Sqrt[Sum[u[[k]] Conjugate[u[[k]]],{k,1,Length[u]}]]