Visualization of Vectors and Span in R2 and R3Worksheet by Michael K. May, S.J., revised by Russell Blyth.Expanded by Steve Mills, Hugh Sanders and Regina Souza --- Draft 6restart: with(LinearAlgebra): with(plottools): with(plots):List of things to be fixed: 1. The numbering of the formulas does not match the numbering of the sections 2. The formulas that are copied and pasted not always adhere to the Maple Input convention (uses Math 2D at times) (See command for plotting w2 multiplied by t (first demo of multiplication by scalars in R^3) - and - cannot be edited (there seemly is some authomatic shift-returns in between symbols) 3. Need an intelligent way of defining the (+,+) grid of the linear combination of two vectors in R^3 (or R^2) using integers from 0 to 5, say. (Messy attempt below.) 4. 5. You name it!
<Text-field style="Heading 1" layout="Heading 1">Outline</Text-field>The basic objectives are:1) Learn the basic mechanics of entering vectors, and producing linear combinations with either addition or scalar multiplication.2) Learn to plot a set of vectors in R2 and R3.3) Visualize the effects of multiplication by scalar and of addition of vectors in R2 and R34) Using a random number generator, see what typical linear combinations of a pair of vectors look like.5) See the effect of linear transformations on the linear combination of vectors6) Apply these concepts to understand and visualize the parametric description of a line and a plane in R3
<Text-field style="Heading 1" layout="Heading 1">1. Vectors in R<Font superscript="true">2</Font> and R<Font superscript="true">3</Font>:</Text-field>The easiest way to enter a vector in R2 and R3 is as a list with angle brackets. In Maple you separate the coordinates with commas for a column vector, and with vertical bars ( | ) for a row vector. The whole vector is surrounded the with angle brackets ( < and >).
Mentally check the computations Maple is doing.Notice that vectors need to have the same length before we can add them:<1,2> + <3,4,5>;We can also enter vectors in Maple with the Vector command, which is part of the LinearAlgebra package.
Mentally check the computations Maple is doing.
<Text-field style="Heading 2" layout="Heading 2">Exercises:</Text-field>1.1) Use the last 4 digits of your telephone number to create two vectors u1 and u2 in R2. Use Maple to compute the linear combination 2*u1 + 3*u2. (Be sure to label answers to all exercises. You can either add a comment like "The answer is ..." to the Maple worksheet, or write a comment on your printout.)2.1) Pick six integers from -10 to 10 (repetitions are allowed) to create two distinct nonzero vectors z1 and z2 in R3. Use Maple to compute 1.0*z1 + 2.0*z2. Compare this to z1+2*z2.
<Text-field style="Heading 1" layout="Heading 1">2. Plotting Lists of Points:</Text-field>We plot points representing vectors with the command pointplot, which is part of the plot package.
Notice that we can plot either a set of points (sets are enclosed in curly braces and are unordered) or a list of points (lists are ordered and enclosed in square brackets). When plotting, you may want to use the view option to specify the viewing window of the plot. For the two plots above, letting x and y both range from -5 to 5 is convenient. You can also specify a symbol size to make the points easier to see.
If the vectors are in R3 instead of R2, we use the command pointplot3dUnfortunately, the default option for 3-dimensional plots in Maple is to hide the axes. This can be fixed by either clicking once on the 3-D plot above and then clicking on the icon for normal axes or by using the axes=normal option. Once again there is a view option for these 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Click on the graph and rotate the plot to get a good idea of the location of the two points.To help visualize the point in space, it might be helpful to plot the dashed lines (the red and green dashlines indicate the projection of the points on the xy-coordinate 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can think of the tip of the red and green segments as the points, and of the arrows as the vectors. Click on the graph and rotate the plot to get a good idea of the location of the two points.After you click on the graph, try typing \316\270=\342\210\222160 [Enter] and \317\225=60 [Enter] on the boxes (located at the top left of the tool bar). Which angles \316\270 and \317\225 give you a good view of these vectors?
<Text-field style="Heading 2" layout="Heading 2">Exercises:</Text-field>2.1) Plot the points [1, 1], [2, -2], [-3, 3], and [4, -4] all on the same graph.2.2) Using the points z1 and z2 you defined in Exercise 2 above, plot z1, z2, z1 + z2, and 2*z1 - z2 all on the same graph.2.3) Include the dashed lines and vectors to the pictures of z1, z2 and z1+z2. Choose angles \316\270 and \317\225 that give you a good view of these three vectors.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
<Text-field style="Heading 1" layout="Heading 1">3. Visualizing operations with vectors:</Text-field>Let us start by visualizing the effect of multiplying vectors by scalars. Let us use v1=[1 1] defined above.
Using the sequence command, we will graph the multiples of v1 in the window below, using scalars `t` from 0 to 5.What happens if you use scalars `s` from \342\210\2225 to 0? Modify the commands and plot the points in the window below.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 := {seq(t*v1,t=0..5)}:
pointplot(v1MultByt,view=[-5..5,-5..5]);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
What is the pattern?Is it any different in 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
Using the sequence command, we will graph the multiples of w2=[-1 3 2] in the window below, using scalars `t` from 0 to 5.What happens if you use scalars `s` from \342\210\2225 to 0? Modify the commands and plot the points in the window below.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
What is the pattern?Now let us see the geometric properties of addition of vectors.In will use v1=[1,1] and v2=[1,3] redefined below.Here we will include the arrows (and segments parallel to them) to make clear what is happening geometrically.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
vectorv1 := (plots[arrow](v1,color=red, shape=arrow,thickness=2)):
vectorv2 := (plots[arrow](v2,color=green, shape=arrow,thickness=2)):
sumv1v2 := (plots[arrow](v1+v2,color=blue, shape=arrow,thickness=2)):
translv2:= line(v1,v1+v2, color = green, thickness = 1, linestyle = DASH):
translv1:= line(v2,v1+v2, color = red, thickness = 1, linestyle = DASH):
display([vectorv1, vectorv2, sumv1v2,translv1,translv2],view=[-5..5,-5..5]);Make the analogous picture for v1-v2 (rename new lines, so you can plot the other plot and the new one side by 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 the previous picture and your picture side by side
Let us see what happens in R^3:Let us see what happens in R^3:
We will use w1=[1,1,1] and w2=[-1,3,2] redefined below.Here we will include the arrows (and segments parallel to them) to make clear what is happening geometrically.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
vectorw1 := (plots[arrow](w1,color=red, shape=arrow,thickness=2)):
vectorw2 := (plots[arrow](w2,color=green, shape=arrow,thickness=2)):
sumw1w2 := (plots[arrow](w1+w2,color=blue, shape=arrow,thickness=2)):
translw2:= line(w1,w1+w2, color = green, thickness = 1, linestyle = DASH):
translw1:= line(w2,w1+w2, color = red, thickness = 1, linestyle = DASH):
display([vectorw1, vectorw2, sumw1w2,translw1,translw2],axes=normal,view=[-6..6,-6..6,-6..6]);Modify the code to get the analogous picture for v1-v2 (rename new lines, so you can plot and compare v1+v2 with v1-v2). 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, in the same way we played with the scalars when trying to visualize the effect of multiplication of a vector by a scalar, we are going to vary the scalars and then add the vectors as in w = s*w1+t*w2. The vector w is said to be a linear combination of w1 and w2.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tw1 := {seq(t*w1,t=0..5)}:
tw2 := {seq(t*w2,t=0..5)}:
tw1w2 := {seq(t*w1+w2,t=0..5)}:
tw12w2 := {seq(t*w1+2*w2,t=0..5)}:
tw13w2 := {seq(t*w1+3*w2,t=0..5)}:
tw14w2 := {seq(t*w1+4*w2,t=0..5)}:
tw15w2 := {seq(t*w1+5*w2,t=0..5)}:
tw1grid := pointplot3d(tw1, axes=normal,view = [-30..30, -30..30, -30..30],color=red, symbol=diamond, symbolsize=5):
tw2grid := pointplot3d(tw2, axes=normal,view = [-30..30, -30..30, -30..30],color=green, symbol=diamond, symbolsize=5):
tw1w2grid := pointplot3d(tw1w2, axes=normal,view = [-30..30, -30..30, -30..30],color=violet, symbol=diamond, symbolsize=5):
tw12w2grid := pointplot3d(tw12w2, axes=normal,view = [-30..30, -30..30, -30..30],color=orange, symbol=diamond, symbolsize=5):
tw13w2grid := pointplot3d(tw13w2, axes=normal,view = [-30..30, -30..30, -30..30],color=pink, symbol=diamond, symbolsize=5):
tw14w2grid := pointplot3d(tw14w2, axes=normal,view = [-30..30, -30..30, -30..30],color=brown, symbol=diamond, symbolsize=5):
tw15w2grid := pointplot3d(tw15w2, axes=normal,view = [-30..30, -30..30, -30..30],color=coral, symbol=diamond, symbolsize=5):
display([vectorw1, vectorw2, sumw1w2,translw1,translw2,w1grid,w2grid,w1w2grid,tw1w2grid,tw12w2grid,tw13w2grid,tw13w2grid,tw14w2grid,tw15w2grid],axes=normal,view=[-1..10,-1..10,-1..10]);
Remarks:In R3: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:
<Text-field style="Heading 2" layout="Heading 2">Exercises:</Text-field>3.1) Let vector v1=[-5 4]. Choose a scalar \316\261 such that the vector v2=\316\261*v1 is twice as long as v1. Choose a scalar \316\262 such that v3=\316\262*v1 has the same length as v1 but it is on a different quadrant. Verify the claims about the lengths by using Pythagoras theorem. Plot the vectors v1, v2 and v3 using colorful arrows. 3.2) Addition of vectors in R2.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.3) Multiplication by scalars in R3.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.4) Addition of vectors in R3.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
<Text-field style="Heading 1" layout="Heading 1">4. Random Number Generators and Long Lists:(nothing new)</Text-field>It is useful to be able to generate random vectors and matrices. The rand function in Maple returns random integers in a specified range. We can use rand to create functions that produce random 3 digit numbers either from 0 to 1 or from -1 to 1.rand0to1 := rand(0..1000)/1000.0:
randneg1to1 := rand(-1000..1000)/1000.0:With the first of these functions it is easy to produce a list of 10 random linear combinations of the form A*v1+B*v2, where A and B are both between 0 and 1.setofpoints := {seq(rand0to1()*v1+rand0to1()*v2,i=1..10)};
pointplot(setofpoints,view=[-5..5,-5..5]);
<Text-field style="Heading 2" layout="Heading 2">Exercise:</Text-field>4.1) Use the rand0to1 function to create a list of 500 random linear combinations of v1 and v2. (You probably want to end the command with a colon rather than a semicolon so the list is not printed out.) Plot the points in the list and describe the geometric figure that they make. Include the coordinates of the vertices in your description.When we try the same trick with vectors in R3, we find that the points all lie in a plane. To see this, rotate the figure below in such a way that the plane is viewed edge-on - that is, so that it appears to be a line.setofpoints := {seq(rand0to1()*w1+rand0to1()*w2,i=1..500)}:
pointplot3d(setofpoints,view=[-1..1,0..4,0..3], axes=normal);When a 3-D plot is active, you see the present orientation in the second menu bar. 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 gives the angle of view in degrees in the xy-plane around from the positive x-axis, while 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 gives the view angle down from vertical. The default view orientation is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2OVEmdGhldGFGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dGNy8lKnVuZGVybGluZUdGNy8lKnN1YnNjcmlwdEdGNy8lLHN1cGVyc2NyaXB0R0Y3LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2JhY2tncm91bmRHRkIvJSdvcGFxdWVHRjcvJStleGVjdXRhYmxlR0Y3LyUpcmVhZG9ubHlHRjcvJSljb21wb3NlZEdGNy8lKmNvbnZlcnRlZEdGNy8lK2ltc2VsZWN0ZWRHRjcvJSxwbGFjZWhvbGRlckdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lKm1hdGhjb2xvckdGQi8lL21hdGhiYWNrZ3JvdW5kR0ZCLyUrZm9udGZhbWlseUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSltYXRoc2l6ZUdGNA== = 45 and 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 = 45. Trial and error rotations of the figure above show that an orientation of [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, 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] = [17,65] views the plane containing all the points on edge, while an orientation of [-11, 17] looks at that plane from the top so that the set of points looks like a figure in a plane.
<Text-field style="Heading 2" layout="Heading 2">Exercise:</Text-field>4.2)
<Text-field style="Heading 1" layout="Heading 1">5. The effect of Linear Transformations on Linear Combinations (Steve mailed that part - Mike has given some further suggestions)</Text-field>
<Text-field style="Heading 1" layout="Heading 1">6. Parametric Equations for Lines and Planes:</Text-field>In this part of the worksheet we are going to look at parametric equations for lines and planes in R3.First, for parametric equations of a line, let's enter a vector v in R3 that will be in the direction of the line and a vector r0 in R3 that will be pointing from the origin to a point on the line.So, first enter the vector v:(Here we'll have the student enter a vector, such as, v = <1, 4, -2>)v := <1, 4, -2>;(We will define a to be the x component of v, b to be the y component of v and c to be the z component of v.So, in this example a = 1, b = 4, and c = -2)a := v[1];
b := v[2];
c := v[3];Next, enter the vector r0:(Here we'll have the student enter a vector, such as, r0 = <5, 1, 3>)r0 := <5, 1, 3>;(We will define x0 to be the x component of r0 , y0 to be the y component of r0 , z0 to be the z component of r0So, in this example x0 = 5, y0 = 1, and z0 = 3.x0 := r0[1];
y0 := r0[2];
z0 := r0[3];The parametric equations for this line are the following:x = x0 + atx := x0 + a*t;y = y0 + bty := y0 + b*t;z = z0 + ct)z := z0 + c*t;The graph of this line is as follows:(Graph the vector function:r = r0 + tv)r := r0 + t*v;
spacecurve(r,t=-5..5,axes=boxed, thickness=5,color=red);(Here would be the Maple code for plotting this line.)Now, let's look at the equation of a plane in R3. Let's use the two vectors that we used to obtain the parametric equations of the line. So, for our plane, lets let the vector r0 point from the origin to a point on the plane and the vector r be a vector that is normal to the plane.The equation of the plane that meets these conditions is the following:a(x - x0) + b(y - y0) + c(z - z0) = 0(For our example, the equation would be 1(x - 5) + 4(y - 1) -2(z - 3) = 0)(Graph this function.)(Here would be the Maple code for plotting this plane.)
<Text-field style="Heading 2" layout="Heading 2">Exercise:</Text-field>6.1) 6.2)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