Chapter 6 - Basic Vector Algebra Topics

Section 6.1 - Vectors

Example 6.1.1

P = {-7, -19}; Q = {20, 14};

V={13,0};W={9,15};

TwoV=2 V

TwoV=2 V

NegThreeW=-3 W

ArrowPlots=Graphics[{Arrowheads[.10],Thickness[.010],Blue,Arrow[{{0,0},W}],Blue,Arrow[{P,W+P}],Red,Arrow[{Q,V+Q}],Red,Arrow[{P,TwoV+P}],Blue,Arrow[{Q,NegThreeW+Q}]}];

TxtPlot=Graphics[{Black,Text["W",{2,7}],Text["V",{25,12}],Text["2V",{6,-22}],Text["-3W",{14,-2}],Text["W",{-4,-10}]}];

Show[ArrowPlots,TxtPlot]

Example 6.1.2

V={-10,17};W={35,8};

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

TxtPlots=Graphics[{Black,Text["V",{-4,11}],Text["W",{16,1}],Text["V+W",{16,11}]}];

LinePlots=Graphics[{Thickness[.005],Blue,Line[{V,V+W}],Blue,Line[{W,V+W}]}];

Show[ArrowPlots,TxtPlots,LinePlots]

Example 6.1.3

ArrowPlots2=Graphics[{Arrowheads[.11],Thickness[.010],Black,Arrow[{{0,0},W}],Black,Arrow[{{0,0},V}],Black,Arrow[{{0,0},-W}],Red,Arrow[{W,W+(V-W)}],Red,Arrow[{{0,0},V-W}]}];

TxtPlots2=Graphics[{Black,Text["V",{-0.5,7}],Text["W",{18,2.5}],Text[“-W”,{-16,-6}],Text["V-W",{18,14}],Text["V-W",{-20,6}]}];

LinePlots2=Graphics[{Thickness[.005],Blue,Line[{V,V-W}],Blue,Line[{V-W,-W}]}];

Show[ArrowPlots2,LinePlots2,TxtPlots2,AspectRatio-> 1]

Example 6.1.4

2.5 V+3.7 W

ArrowPlots3=Graphics[{Arrowheads[.07],Thickness[.010],Black,Arrow[{{0,0},V}],Black,Arrow[{{0,0},2.5 V}],Black,Arrow[{{0,0},W}],Black,Arrow[{{0,0},3.7 W}],Red,Arrow[{{0,0},2.5 V+3.7 W}]}];

TxtPlots3=Graphics[{Black,Text["V",{-7.5,2}],Text["W",{18,-1}],Text["2.5V",{-4,25}],Text["3.7W",{100,15}],Text["2.5V+3.7W",{75,37}]}];

LinePlots3=Graphics[{Thickness[.005],Blue,Line[{2.5 V,2.5 V+3.7 W}],Blue,Line[{3.7 W,2.5 V+3.7 W}]}];

Show[ArrowPlots3,LinePlots3,TxtPlots3,PlotRange-> {{-27,133},{-10,75}},Axes-> True,AxesOrigin-> {-27,-10}]

Example 6.1.5

V={5,3};VI={1,0};V5I={5,0};VJ={0,1};V3J={0,3};

ArrowPlots4=Graphics[{Arrowheads[.07],Thickness[.010],Black,Arrow[{{0,0},VI}],Black,Arrow[{{0,0},V5I}],Black,Arrow[{{0,0},VJ}],Black,Arrow[{{0,0},V3J}],Red,Arrow[{{0,0},V}]}];

TxtPlots4=Graphics[{Black,Text["J",{-0.25,0.25}],Text["I",{0.5,-0.25}],Text["3J",{-0.25,2.5}],Text["5I",{4,-0.25}],Text["V=5I+3J",{3,1.25}]}];

Show[ArrowPlots4,TxtPlots4,PlotRange-> {{-0.5,5},{-0.5,3}},Axes-> True,AxesOrigin-> {-0.5,-0.5}]

Example 6.1.6

V={4,-7,-3};VI={1,0,0};VJ={0,1,0};VK={0,0,1};

V4I={4,0,0};VMinus7J={0,-7,0};VMinus3K={0,0,-3};

ArrowPlots5=Graphics3D[{Arrowheads[.05],Thickness[.010],Black,Arrow[{{0,0,0},VI}],Black,Arrow[{{0,0,0},VJ}],Black,Arrow[{{0,0,0},VK}],Blue,Arrow[{{0,0,0},V4I}],Blue,Arrow[{{0,0,0},VMinus7J}],Blue,Arrow[{{0,0,0},VMinus3K}],Red,Arrow[{{0,0,0},V}]}];

TxtPlots5=Graphics3D[{Black,Text["I",{1,0,0}],Text["J",{0,1,0}],Text["K",{0,0,1}],Text["4I",{4,0,0}],Text["-7J",{0,-7,0}],Text["-3K",{0,0,-3}],Text["V=4I-7J-3K",{4,-7,-3}]}];

Show[ArrowPlots5,TxtPlots5,Axes-> True,PlotRange-> {{-1,6},{-8,2},{-4,2}},AspectRatio-> 1,ViewPoint-> {8,-2,5}]

Example 6.1.7

P={-8,11};Q={7,4};

V=Q-P

ArrowPlots6=Graphics[{Arrowheads[.07],Thickness[.010],Black,Arrow[{{0,0},P}],Black,Arrow[{{0,0},Q}],Red,Arrow[{P,Q}],Blue,Arrow[{{0,0},V}]}];

TxtPlots6=Graphics[{Black,Text["P",{-5,5}],Text["Q",{4.5,1}],Text["PQ",{0,9}],Text["V",{10,-3}]}];

Show[ArrowPlots6,TxtPlots6,PlotRange-> {{-10,17},{-9,13}},Axes-> True,AxesOrigin-> {-10,-9}]

Commands after Example 6.1.7

V={-10,17};W={35,8};

ArrowPlots7=Graphics[{Arrowheads[.07],Thickness[.010],Black,Arrow[{{0,0},V}],Blue,Arrow[{{0,0},W}]}];

ArcPlot=Graphics[{Red,Thickness[0.015],Circle[{0,0},7,{ArcTan[8/35],π-ArcTan[17/10]}]}];

TxtPlots7=Graphics[{Black,Text["θ", {1,3}],Text["V",{-5,12}],Text["W", {20,2.5}]}];

Show[ArcPlot,ArrowPlots7,TxtPlots7,Axes->True,AxesOrigin->{-13,0}]