(*^
::[	Information =

	"This is a Mathematica Notebook file.  It contains ASCII text, and can be
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	The line below identifies what version of Mathematica created this file,
	but it can be opened using any other version as well.";

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Your Assignment:
1) Execute the entire notebook.  Experiment if you like. 
2) Do Problem 21  & Problem 51 as best you can.
3) Hand in Prob.21 and Prob.51
;[s]
2:0,0;16,1;154,-1;
2:1,19,14,Times,1,18,0,0,0;1,16,12,Times,1,14,0,0,0;
:[font = section; inactive; initialization; Cclosed; preserveAspect; startGroup]
Initialization: 
(Don't read this; just execute it)
(Click on the outer bracket--far right--and hit the ENTER key.
;[s]
2:0,0;17,1;115,-1;
2:1,19,14,Times,1,18,0,0,0;1,16,12,Times,1,14,0,0,0;
:[font = text; inactive; initialization; preserveAspect]
Define the cross product:
:[font = input; initialization; preserveAspect]
*)
Needs["LinearAlgebra`CrossProduct`"]
(*
:[font = input; initialization; preserveAspect]
*)
Off[ General::spell ];
Off[ General::spell1];
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear["Global`*"];

VectorPlot[list__List?VectorQ]:= VectorPlot[{list}];

VectorPlot[list_List?MatrixQ]:= 
Module[{xlist,length,totList},
 xlist=Join[{{0,0}},list];
 length=Length[xlist];
 totList=Table[ Sum[ xlist[[i]] ,{i,1,j}],{j,1,length}];
 Return[
  Show[Graphics[{Thickness[0.015],Line[totList]}
          ,{GridLines->Automatic
           ,Axes->True}]]
 ]
];
(*
:[font = message; inactive; preserveAspect; startGroup]
SetDelayed::write: Tag VectorPlot in VectorPlot[(list__List)?VectorQ] is Protected.
:[font = message; inactive; preserveAspect; endGroup; endGroup]
SetDelayed::write: Tag VectorPlot in VectorPlot[(list_List)?MatrixQ] is Protected.
:[font = input; initialization; preserveAspect; startGroup]
*)
VectorPlot3D[list__List?VectorQ]:= VectorPlot3D[{list}];

VectorPlot3D[list_List?MatrixQ]:= 
Module[{xlist,length,totList},
 xlist=Join[{{0,0,0}},list];
 length=Length[xlist];
 totList=Table[ Sum[ xlist[[i]] ,{i,1,j}],{j,1,length}];
 Return[
  Show[Graphics3D[{Thickness[0.015],Line[totList]}
          ,{ FaceGrids->{{0,0,-1},{0,1,0},{-1,0,0}}
            ,BoxRatios->{1,1,1}
            ,Axes->True}]]
 ]
];
(*
:[font = message; inactive; preserveAspect; startGroup]
SetDelayed::write: Tag VectorPlot3D in VectorPlot3D[(list__List)?VectorQ] is Protected.
:[font = message; inactive; preserveAspect; endGroup; endGroup]
SetDelayed::write: Tag VectorPlot3D in VectorPlot3D[(list_List)?MatrixQ] is Protected.
:[font = input; initialization; preserveAspect; startGroup]
*)

VectorLength[vector_List?VectorQ]:=Sqrt[vector.vector]
(*
:[font = message; inactive; preserveAspect; startGroup]
SetDelayed::write: Tag VectorLength in VectorLength[(vector_List)?VectorQ] is Protected.
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
$Failed
;[o]
$Failed
:[font = input; initialization; preserveAspect; endGroup]
*)
Protect[{VectorPlot,VectorPlot3D,VectorLength}];
Off[Clear::wrsym];
(*
:[font = section; inactive; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
2-Dimensional Vectors: Warmup
:[font = text; inactive; preserveAspect]
Define three  2-Dimensional vectors:
:[font = input; preserveAspect]

Clear["Global`*"];

v1={1,0};
v2={1,1};
v3={-3,1};
:[font = input; preserveAspect]
VectorPlot[v1,v2,v3];

:[font = text; inactive; preserveAspect]
Find the length of the sum.  Does this match the plot???
:[font = input; preserveAspect]
VectorLength[v1+v2+v3]
:[font = text; inactive; preserveAspect; endGroup]
Define 3 2-Dimensional vectors:
:[font = section; inactive; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
3-Dimensional Vectors:  Warmup
:[font = input; preserveAspect]
Clear["Global`*"];

VectorPlot3D[{1,1,1},{1,-1,-1},{1,2,3}];

:[font = text; inactive; preserveAspect]
Find the length of the sum.  Does this match the plot???
:[font = input; preserveAspect; endGroup]
VectorLength[{1,1,1}+{1,-1,-1}+{1,2,3}]

:[font = section; inactive; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Ch 3: Problem 5
:[font = text; inactive; preserveAspect]
Let us define the 3 vectors needed. We have to work a little bit for v3. Note how I convert from Radians to Degrees. Sorry, but the natural unit for trigonometric functions is Radians. 
:[font = input; preserveAspect]
Clear["Global`*"];

v1={50,0};
v2={0,30};
v3={25 Cos[60/Degree], 25 Sin[60/Degree]} //N
:[font = text; inactive; preserveAspect]
Let us look at the path that the car took.  Can you estimate from the figure the length of the final displacement vector???
:[font = input; preserveAspect]
VectorPlot[v1,v2,v3];
:[font = text; inactive; preserveAspect]
Let 's compute the total length. Does it match the figure???
:[font = input; preserveAspect]
v4=v1+v2+v3;
{v4x,v4y}=v4
:[font = input; preserveAspect]
VectorLength[v1+v2+v3]
:[font = text; inactive; preserveAspect]
Let 's compute the angle. Note I have to convert from Radians to Degrees.
:[font = input; preserveAspect; endGroup]
theta=ArcTan[v4y/v4x] /Degree  //N
:[font = section; inactive; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Ch 3: Problem 9
:[font = text; inactive; preserveAspect]
Here, we get the magnitude and direction. Again, I have to convert 
:[font = input; preserveAspect]
Clear["Global`*"];

r=7.3;
theta=250 Degree
:[font = input; preserveAspect]
a={r Cos[theta], r Sin[theta]} //N
:[font = input; preserveAspect; endGroup]
{ax,ay}=a
:[font = section; inactive; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Ch 3: Problem 21
:[font = text; inactive; preserveAspect; endGroup]
This is easy with Mathematica. Give it a try.

Find the vector sum of  c={7.4, -3.8, -6.1} and d={4.4, -2.0, 3.3} where r=c+d
;[s]
3:0,0;47,1;125,0;126,-1;
2:2,13,9,Times,0,12,0,0,0;1,16,12,Times,0,14,0,0,0;
:[font = section; inactive; pageBreak; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Ch 3: Problem 22
:[font = text; inactive; preserveAspect]
Let's define the necessary vectors.
:[font = input; preserveAspect]
Clear["Global`*"];

a={4,3};
b={-13,7};
c=a+b
:[font = text; inactive; preserveAspect]
Let's see what they look like. Can you guess at the angle and magnitude of the total vector c???
:[font = input; preserveAspect]
VectorPlot[a,b];
:[font = text; inactive; preserveAspect]
Let's calculate the length.
:[font = input; preserveAspect]
VectorLength[c] //N
:[font = text; inactive; preserveAspect]
To find the angle, we need the x and y components of c. Here is an easy way to do it.
:[font = input; preserveAspect]
{cx,cy}=c
:[font = text; inactive; preserveAspect]
The Tangent is the ratio of y/x.
:[font = input; preserveAspect]
tanTheta=cy/cx
:[font = text; inactive; preserveAspect]
Now we take the ArcTan (Inverse Tangent) and convert to degrees.
:[font = input; preserveAspect]
theta=ArcTan[tanTheta] /Degree //N
:[font = text; inactive; preserveAspect]
Note that we get the wrong Quadrant by using the math. Use the figure to see that the correct answer is:
:[font = input; preserveAspect; endGroup]
theta+180 
:[font = section; inactive; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Ch 3: Problem 48
:[font = text; inactive; preserveAspect]
Let's define the necessary vectors.
:[font = input; preserveAspect]
Clear["Global`*"];

a={3,3,3};
b={2,1,3};
:[font = text; inactive; preserveAspect]
Let's calculate the length.
:[font = input; preserveAspect]
alength=VectorLength[a]
:[font = text; inactive; preserveAspect]
Let's calculate the length.
:[font = input; preserveAspect]
blength=VectorLength[b]
:[font = text; inactive; preserveAspect]
The cosine is given by:  (Note the DOT product between the vectors a and b.)
:[font = input; preserveAspect]
cosTheta=a.b/(VectorLength[a] VectorLength[b])
:[font = text; inactive; preserveAspect]
Now we take the ArcCos (Inverse Cosine) and convert to degrees.
:[font = input; preserveAspect]
theta=ArcCos[cosTheta] /Degree //N
:[font = input; preserveAspect; endGroup]
Show[ VectorPlot3D[a],VectorPlot3D[b] ];
:[font = section; inactive; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Ch 3: Problem 51
:[font = text; inactive; preserveAspect]
Let's define the necessary vectors.  Note that I make both a and b 3-dimensinal vectors by including a zero term for the z component. We only know how to compute a cross product in 3-dimensions. 
:[font = input; preserveAspect]
Clear["Global`*"];

a={3,5,0};
b={2,4,0};
:[font = subsection; inactive; preserveAspect]
Part a)
:[font = text; inactive; preserveAspect]
The Cross product for vectors was defined by loading in the package LinearAlgebra`CrossProduct  with the command Needs["LinearAlgebra`CrossProduct`"]. Let us see what Mathematica knows about Cross.
;[s]
3:0,0;167,1;178,0;198,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;
:[font = input; preserveAspect; startGroup]
?Cross
:[font = info; inactive; preserveAspect; endGroup]
Cross[vec1, vec2] gives the vector product of the 3-vectors vec1 and vec2.
:[font = text; inactive; preserveAspect]
Let us compute axb.
:[font = input; preserveAspect]
Cross[a,b]
:[font = text; inactive; preserveAspect]
axb should be the negative of bxa. Is this true???
:[font = input; preserveAspect]
Cross[b,a]
:[font = text; inactive; preserveAspect]
If I had to do it by hand, I would first define the matrix:
:[font = input; preserveAspect]
matrix= {{vx,vy,vz},a,b};
matrix //MatrixForm
:[font = text; inactive; preserveAspect]
and then take the Det (determinant) of this matrix.
:[font = input; preserveAspect]
Det[matrix]
:[font = text; inactive; preserveAspect]
We get the same answer.
:[font = subsection; inactive; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
You do part b) and c). (They are very easy.)
:[font = text; inactive; preserveAspect; fontSize = 14]

Two vectors are given by: a={3,5,0} and b={2,4,0}
Find:      a.b      (the dot product)
            (a+b).b
Note, we use the period "." for the dot product. 
Or you can use Dot[a,b]
;[s]
5:0,0;61,1;67,0;102,1;109,0;183,-1;
2:3,16,12,Times,0,14,0,0,0;2,16,12,Times,1,14,0,0,0;
:[font = input; preserveAspect; startGroup]
?.
:[font = info; inactive; preserveAspect; endGroup; endGroup; endGroup]
a.b.c or Dot[a, b, c] gives products of vectors, matrices and tensors.
^*)