Vectors in Space. Standard Graphing

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1 Vectors in Space O restart:with(plots);with(vectorcalculus): animate, animated, animatecurve, arrow, changecoords, complexplot, complexplotd, conformal, conformald, contourplot, contourplotd, coordplot, coordplotd, densitplot, displa, dualaxisplot, fieldplot, fieldplotd, gradplot, gradplotd, graphplotd, implicitplot, implicitplotd, inequal, interactive, interactiveparams, intersectplot, listcontplot, listcontplotd, listdensitplot, listplot, listplotd, loglogplot, logplot, matrixplot, multiple, odeplot, pareto, plotcompare, pointplot, pointplotd, polarplot, polgonplot, polgonplotd, polhedra_supported, polhedraplot, rootlocus, semilogplot, setcolors, setoptions, setoptionsd, spacecurve, sparsematrixplot, surfdata, textplot, textplotd, tubeplot Standard Graphing To displa the NORMAL coordinate axes, let's graph the points (,,) and (,,6), and join them with a line. O p:=pointplotd({[,,],[,,6]},axes=normal,labels=[x,,], connect=true,color=blue,thickness=): p:=textplotd([,,,"(,,6)",font=[times,bold,],color= magenta]): displa(p,p);

2 6.. x.. (,,6). We can also view our graph within a box, using the option "axes=boxed." O p:=pointplotd({[,,],[,,6]},axes=boxed,labels=[x,,], connect=true,color=blue,thickness=): p:=textplotd([.,,,"(,,6)",font=[times,bold,],color= magenta]): displa(p,p);

3 6 (,,6).... x. The perspective seems all wrong in both cases. It is the default orientation [, ]=[, ]. Let's tr an alternate orientation of [, ]=[,7]. O p:=pointplotd({[,,],[,,6]},axes=normal,labels=[x,,], connect=true,color=blue,thickness=,orientation=[,7]): O p:=plotd([*s,*s,],s=..,t=..,axes=normal,labels=[x,,], thickness=,orientation=[,7]): O p:=plotd([*s,*s,6],s=..,t=..,axes=normal,labels=[x,,], thickness=,orientation=[,7]): O p:=plotd([,,s],s=..6,t=..,axes=normal,labels=[x,,], thickness=,orientation=[,7]): p:=textplotd([,,7,"(,,6)",font=[times,bold,],color= magenta]): O displa(p,p,p,p,p);

4 7 6 (,,6).. x... This is better. An orientation of [, ]=[,] has the positive axis pointing out perpendicularl from the screen, the positive x axis pointing downward, and the positive axis pointing to the right. A of moves the x- plane degrees clockwise, and a of 7 tilts the positive -axis so that it makes an angle of (9-7) degrees with the screen. We will use the setoptionsd command to set d graphing options that we want for all d graphs until Maple is restarted. These no longer need to be explicitl stated as the were above. O setoptionsd(axes=normal,labels=[x,,],orientation=[,7]); We see we now get the same graph as above without needing to tpe in so man options. O p:=pointplotd({[,,],[,,6]},connect=true,color=blue, thickness=): O p:=plotd([*s,*s,],s=..,t=..,thickness=): O p:=plotd([*s,*s,6],s=..,t=..,thickness=): O p:=plotd([,,s],s=..6,t=..,thickness=): p:=textplotd([,,7,"(,,6)",font=[times,bold,],color= magenta]): O displa(p,p,p,p,p);

5 7 6 (,,6).. x... Some people prefer a boxed axis sstem. Let's look at the above graph with boxed axes. O p:=pointplotd({[,,],[,,6]},connect=true,color=blue, thickness=): O p:=plotd([*s,*s,],s=..,t=..,thickness=): O p:=plotd([*s,*s,6],s=..,t=..,thickness=): O p:=plotd([,,s],s=..6,t=..,thickness=): p:=textplotd([,,7,"(,,6)",font=[times,bold,],color= magenta]): O displa(p,p,p,p,p,axes=boxed);

6 7 6 (,,6)... x.. Notice how the default axes option was overridden b the option "axes=boxed" in the displa command. Another stle that can be used is a framed axis sstem. O p:=pointplotd({[,,],[,,6]},connect=true,color=blue, thickness=): O p:=plotd([*s,*s,],s=..,t=..,thickness=): O p:=plotd([*s,*s,6],s=..,t=..,thickness=): O p:=plotd([,,s],s=..6,t=..,thickness=): p:=textplotd([,,7,"(,,6)",font=[times,bold,],color= magenta]): O displa(p,p,p,p,p,axes=framed); 6

7 7 6 (,,6)... x.. Now let's follow the coordinates to reach our point [,,6]. Starting at the origin, we fisrt move a distance of in the positive x direction. O p:=pointplotd({[,,],[,,6]},connect=true,color=blue, thickness=): O p:=plotd([*s,,],s=..,t=..,thickness=): p:=textplotd([,.,,"(,,)",font=[times,bold,],color= magenta]): O displa(p,p,p); 7

8 6.. x... (,,) Then we move in the positive direction. O p:=pointplotd({[,,],[,,6]},connect=true,color=blue, thickness=): O p:=plotd([*s,,],s=..,t=..,thickness=): O p:=plotd([,*s,],s=..,t=..,thickness=): p:=textplotd([,.,,"(,,)",font=[times,bold,],color= magenta]): O displa(p,p,p,p); 8

9 6.. x... (,,) Finall, we move 6 in the positive direction. O p:=pointplotd({[,,],[,,6]},connect=true,color=blue, thickness=): O p:=plotd([*s,,],s=..,t=..,thickness=): O p:=plotd([,*s,],s=..,t=..,thickness=): O p:=plotd([,,s],s=..6,t=..,thickness=): p:=textplotd([,,7,"(,,6)",font=[times,bold,],color= magenta]): O displa(p,p,p,p,p); 9

10 7 6 (,,6).. x... Next, we look at the graphs of the three planes =, = K, and =. Note that, for instance, the equation = means x C C = in this -dimensional context. Also, notice that the graph of = is the x- plane. O plotd({-,,},x=-..,=-..);

11 -. x We can see the axes better b removing the grid, using the option "stle=patchnogrid" (instead of the default PATCH). O plotd({-,,},x=-..,=-..,stle=patchnogrid);

12 -. x We can even extend the vertical axis if we wish, using the view option to set the extent of the -axis. O p:=plotd({-,,},x=-..,=-..,view=-6..6,stle= PATCHNOGRID): p:=textplotd([,,,"x- plane",font=[times,bold,],color= magenta]): displa(p,p);

13 x x- plane Next, let's graph the x- plane using parametric plotting and using the view option to set a viewing window and "scaling = CONSTRAINED" to have equal scales on each axis. O p:=plotd([x,,],x=-6..6,=-6..6,view=[-6..6,-6..6,-6..6], stle=patchnogrid,scaling=constrained): p:=textplotd([,,,"x- plane",font=[times,bold,],color= magenta]): displa(p,p);

14 x x- plane. -6. Finall, we graph the - plane. O p:=plotd([,,],=-6..6,=-6..6,view=[-6..6,-6..6,-6..6], stle=patchnogrid,scaling=constrained): p:=textplotd([,.,,"- plane",font=[times,bold,],color= magenta]): displa(p,p);

15 plane -6. x Defining free vectors (arrows), rooted vectors, and position vectors in dimensions. So that we can enter vectors using standard vector notation, we set BasisFormat to false. O BasisFormat(false): We can enter free vectors as either column (the default) or row vectors. There are two was to enter column vectors. O v:=<,-,7>; v := K 7 O v:=vector([,-,]); v := K We enter row vectors as follows. O v:=< 7>;

16 v := 7 We use the About statement to gain information about these vectors. O About(v);About(v);About(v); Tpe: Free Vector Components:, K, 7 Coordinates: cartesian Tpe: Free Vector Components:, K, Coordinates: cartesian Tpe: Free Vector Components:,, 7 Coordinates: cartesian Free vectors are plotted using PlotVector the command. O p:=plotvector(v,view=[-..,-..,-7..7]): p:=textplotd([,-,,"<,-,7>",font=[times,bold,],color= green]): displa(p,p); 6

17 . <,-,7> x We can adjust the shape of the arrow b using parameters. O p:=plotvector(v,view=[-..,-..,-7..7],width=.,head_width=.,head_length=.,color=blue): p:=textplotd([,-,6,"<,-,7>",font=[times,bold,],color= green]): displa(p,p); 7

18 <,-,7> x If head_length is not given, the head is about the length of the arrow. O p:=plotvector(v,view=[-..,-..,-7..7],width=.,head_width=.,color=can): p:=textplotd([,-,7,"<,-,7>",font=[times,bold,],color= green]): displa(p,p); 8

19 <,-,7> x A rooted vector is an arrow whose tail is fixed to a point in the plane. For instance, place the tail of v at the point (-,-,). O p:=plotvector(<,-,>,v,view=[-..,-..8,-..],width=.,head_width=.,head_length=.,color=green): p:=textplotd([,-8,7,"<,-,7>",font=[times,bold,],color= green]): displa(p,p); 9

20 . 9. <,-,7> x We define and plot a rooted vector using RootedVector as follows. O v:=rootedvector(root=[-,-,-],[,,7]); v := 7 O O About(v); Tpe: Rooted Vector Components:,, 7 Coordinates: cartesian Root Point: K, K, K p:=plotvector(v,view=[-8..8,-..,-..],width=.,head_width=.,head_length=.,color=orange): p:=textplotd([,-,,"<,,7>",font=[times,bold,],color= green]): displa(p,p);

21 <,,7>.. x We define and plot a position vector, a vector with tail at the origin, using PositionVector as follows. O v:=positionvector([,-,]); O O About(v); v := K Tpe: Position Vector Components:, K, Coordinates: cartesian Root Point:,, p:=plotvector(v,view=[-..,-..,-..6],width=.,head_width=.,head_length=.,color=violet): p:=textplotd([,-.,,"<,-,>",font=[times,bold,],color= green]): displa(p,p);

22 . <,-,>.. x Basic Operations for Vectors. O restart:with(plots):with(vectorcalculus):basisformat(false): O setoptionsd(axes=normal,labels=[x,,],orientation=[,7]); We first define six (column) vectors: two free, two rooted, and two position. O v:=<,,>;v:=<-6,,->;v:=rootedvector(root=[-,,],[-,, ]);v:=rootedvector(root=[-,,],[,,-]);v:=positionvector( [,,]);v6:=positionvector([-,,-]); v := v := K6 K

23 v := v := v := v6 := K K K K We use the Norm command to find the norm or magnitude of the six vectors. Since there are several vector norms, with the VectorCalculus package defaulting to the Euclidean (or ) norm and the LinearAlgebra package defaulting the the infinit (or N) norm, we will write Norm(v,) to find the Euclidean norm of a vector v in the usual cases, and use Norm(v,infinit) if we ever want the infinit norm. O Norm(v,);Norm(v,);Norm(v,);Norm(v,);Norm(v,);Norm(v6, ); 8 Vector addition is done b component. We first add two free vectors and then two position vectors. O v:=v+v;about(v);v6:=v+v6;about(v6); K v := 7 K Tpe: Components: Coordinates: Free Vector K, 7, K cartesian

24 v6 := K Tpe: Position Vector Components:,, K Coordinates: cartesian Root Point:,, We see the sums are of the same tpe as the addends. Rooted vectors can onl be added if their roots (points of origin) are the same with the sum having the same root. O v:=v+v;about(v); v := Tpe: Rooted Vector Components:,, Coordinates: cartesian Root Point: K,, What about adding a free or position vector to a rooted vector? O v:=v+v;about(v); K v := 7 O v:=v+v;about(v); Tpe: Rooted Vector Components: K, 7, Coordinates: cartesian Root Point: K,, v := 6 Tpe: Rooted Vector Components:,, 6 Coordinates: cartesian Root Point: K,,

25 We see the sum is a rooted vector with the same root as the rooted addend. What about a free vector with a position vector? O v:=v+v;about(v); 8 v := 6 Tpe: Free Vector Components: 8, 6, Coordinates: cartesian The sum is a free vector. We now wish to visualie vector addition. We must first add the LinearAlgebra subpackage of the Student package. O with(student[linearalgebra]): O p:=vectorsumplot(v,v): p:=textplotd({[,,,"v"],[,,,"v"],[,6,7,"v"],[,6,, "v"],[,9,6,"v=v+v"]},font=[times,bold,],color=black): displa(p,p); The Sum of Vectors.8 v v=v+v.8 v.8 -. x -. v v 6 8 We can see addition b the parallelogram law and the tip-to-tail method are both illustrated for v+

26 v=v. We can also see that vector addition is commutative. We can also use the same diagram to indicate subtraction. O p:=vectorsumplot(v,v): p:=textplotd({[,,,"v"],[,,,"v"],[,6,7,"v=v-v"], [,7,,"v=v-v"],[,,.,"v"]},font=[TIMES,BOLD,],color= black): displa(p,p); The Sum of Vectors v=v-v.8.8 v v.8 -. x.8-. v=v-v v.8.8 The upper triangle illustrates v=v-v, while the lower triangle illustrates v=v-v. We next look at scalar multiplication. We first multipl b. O w:=(/)*v;w:=(/)*v;w:=(/)*v; w := 6

27 w := w := K We view the original vectors in red and the scalar multiples in blue. O p:=plotvector([v,v,v],width=.,head_width=.,head_length=., scaling=constrained,color=red): p:=plotvector([w,w,w],width=.,head_width=.,head_length=., scaling=constrained,color=blue): p6:=textplotd({[,6,,"v"],[,,8,"v"],[.,,,"w"],[,,6, "w"],[,-.,,"v"],[,-,,"w"]},font=[times,bold,],color= black): displa(p,p,p6); 7

28 8 v 6 w v -6 x w w.. v Next we multipl b -. O :=(-)*v;:=(-)*v;:=(-)*v; K6 := K K := := 8 K K6 K K K6 Again we visualie the original vectors in red and the scalar multiples in blue. O p7:=plotvector([v,v,v],width=.,head_width=.,head_length=., scaling=constrained,color=red): p8:=plotvector([,,],width=.,head_width=.,head_length=., scaling=constrained,color=blue): 8

29 p9:=textplotd({[,,,"v"],[,6,9,"v"],[,-7,-,""],[,-6, -,""],[,-,,"v"],[,,-,""]},font=[times,bold,], color=black): displa(p7,p8,p9); 9. v v x v -6. Decomposition in Dimensions O restart:with(plots):with(plottools):with(vectorcalculus) :BasisFormat(false): O setoptionsd(axes=normal,labels=[x,,],orientation=[,7]); We enter the standard vectors. O i:=vector([,,]);j:=vector([,,]);k:=vector([,,]); i := j := 9

30 k := We view the decomposition <,,9> = i + j +9k in three dimensions. O p7:=plotvector([i,j,k],width=.,head_width=.,head_length=., scaling=constrained,color=blue): p8:=plotvector([*i,rootedvector(root=*i,[,,]),rootedvector (root=*i+*j,[,,9]),*i+*j+9*k],width=.,head_width=., head_length=.,scaling=constrained,color=red): p9:=textplotd({[,-.,,"i"],[,,.,"j"],[,-.,,"k"],[,-.,,"i"],[6.,,,"i+j"],[,,9.,"i+j+9k"]},font=[times, BOLD,],color=black): displa(p7,p8,p9); 7. i+j+9k. i. k j. i i+j x.