SharedMath211-18F-PublicShared.sagewsOpen in CoCalc
Author: Paul Zeitz
Description: Now it should work
################################ # # TABLE OF CONTENTS # (You can move to any line by using the "lightning bolt" button to the right of the large 'A') # # 65 Distributive property for dot- and cross-product # 112 Miscellaneous examples, done in class on R 8/28/18 # 125 Plot a plane, given three points on it # 170 Plot a plane, given a normal vector and the distance from the plane to the origin # # #ILLUSTRATION OF DISTRIBUTIVE PROPERTY FOR DOT PRODUCT AND FOR CROSS PRODUCT #In the diagram created, vector A is a unit vector. We draw both parallel and perpendicular projections of vectors B and C onto A. # The main idea is that given any vectors B and A, we can resolve B into two orthogonal components: the PARALLEL component of B onto A is parallel to A, while the # PERPENDICULAR component of B onto A is perpendicular to A and in the same plane as A and B. # The MAGNITUDE of the PARALLEL projection of B onto A is the absolute value of the dot-product of B and A (assume A is a unit vector), # and the PERPENDICULAR projection of B onto A (the actual VECTOR, not the magnitude) is the cross-product of B and A. def perpProj(x,y): #output is the perpendicular component of x wrt y, where #it is understood for convenience that y is a unit vector #and x, y are both vectors return x-y*x.dot_product(y) def parProj(x,y): #output is the parallel component of x wrt y, where y is unit return y*x.dot_product(y) a=vector([0,1,0]) b=vector([-.3,.1,.7]) c=vector([-.7,.5,.3]) t=6 #thickness of some vectors bc=b.cross_product(a) P=plot(a, color='red', thickness=t) P += plot(b, color='blue', thickness=t) P += plot(c, color='black',thickness =t, start=b) #P += plot(c,color='black',start=b) #P += plot(b,color='blue',start=c) P += plot(text3d("B",b*0.7-a*.05,color='blue')) P += plot(text3d("A",a*1.05,color='red')) P += plot(text3d("C",b*1.2+c*0.3,color='black')) P += plot(b+c,color="magenta", thickness=t) P += plot(text3d("B+C", (b+c)*.6, color='magenta')) P += plot(-perpProj(b,a), color='blue',start=b) P += plot(-perpProj(b,a), color='blue',start=b+parProj(c,a)) P += plot(parProj(b,a), color='blue',start=-0.05*perpProj(b,a)) P += plot(parProj(c,a), color='black',start = parProj(b,a)-0.08*perpProj(b,a)) P += plot(parProj(b+c,a), color='magenta',start=-0.12*perpProj(b,a)) P += plot(-perpProj(b+c,a), color='magenta',start=b+c) P += plot(parProj(c,a),color='black',start=b) P += plot(perpProj(c,a), color='black',start=b+parProj(c,a)) #P += plot(parProj(b+c,a), color='magenta',start=perpProj(b+c,a)) P.show(aspect_ratio=1.0,frame=false, spin=5)
3D rendering not yet implemented
78^543
25548726719570371226255187682826237626495114567336561342146579805780338587968682242723434623119650821725595954157695488460462846809201877877101685248835438277545547430259371949782997802506616873836495964781789596164252239566560525969473170257436451198555331307424151262645614190306662519157100743141199967027913417026847216132670063183094986837111669820368791714976148885032137794066342274413941769002125318064629115949358741519464979590531506856333639383376603934970263120108128195945327144359259742403137647398385745075341355975284746043220425048832590748346844105111873022911489985361074484973326385983468023945655325172654194703329836830176481535482759618437408618142114936637666353141144740356606252629412735038494923307954253201828522875665770502526479807331548495980718185340124774147586176082224989136985186394568443152716282859040204105716801734062596717887204195090976233956564686018563963443745498614955635316153284494537511251490152827702742425082397923764331727505659274103850737388252543730256622546808042424369152
plot(sin,0,pi)
%var x,y,z pic=implicit_plot3d(x^2+y^2+z^2-1,(x,-1,0.5),(y,-1,1),(z,-1,1),color='red',mesh=1) pic.show()
3D rendering not yet implemented
#Plot a plane, given three points on the plane # try changing the points defined in line 27 of this #declare variables %var x, y, z #define a function to draw axes. Very useful for understanding plots def axes(size,xcolor='red',ycolor='green',zcolor='blue'): #returns a plot of axis vectors, labeled, with size xSize=size; ySize= size; zSize=size xAxis = plot(vector((xSize,0,0)),color=xcolor,width=3) yAxis = plot(vector((0,ySize,0)),color=ycolor,width=3) zAxis = plot(vector((0,0,zSize)),color=zcolor,width=3) #Labels for axes #set fontsize (change if you need to) fsize=20 xT=text3d("X",(xSize*1.1,0,0),color=xcolor,fontsize=fsize) yT=text3d("Y",(0,ySize*1.1,0),color=ycolor,fontsize=fsize) zT=text3d("Z",(0,0,zSize*1.1),color=zcolor,fontsize=fsize) return xAxis+xT+yAxis+yT+zAxis+zT asize = 16 #axis size psize = 12 #plane size #define the three points a= (0,0,1); b= (0,1,0); c= (1,0,0) #turn em into vectors A = vector(a); B= vector(b); C=vector(c) AB = B-A; AC = C-A N = AB.cross_product(AC) #note the syntax for cross product f = N[0]*x+N[1]*y + N[2]*z-N.dot_product(A) #set this equal to zero to get equation of plane P = plot(axes(asize)) P += implicit_plot3d(f, (x,-psize, psize), (y,-psize, psize), (z, -psize, psize), color='orange', opacity=0.1) P += plot(points([a,b,c], thickness=40, color='black')) P += plot(N, start = a, width = 3, color='black') show('The equation of the plane is $'+latex(f)+'=0$') show('Normal vector is black.') show(P, spin =5)
The equation of the plane is xyz+1=0 -x - y - z + 1 =0
Normal vector is black.
3D rendering not yet implemented
#Plot a plane, given a normal vector and the distance from the origin #declare variables %var x, y, z #define a function to draw axes. Very useful for understanding plots def axes(size,xcolor='red',ycolor='green',zcolor='blue'): #returns a plot of axis vectors, labeled, with size xSize=size; ySize= size; zSize=size xAxis = plot(vector((xSize,0,0)),color=xcolor,width=3) yAxis = plot(vector((0,ySize,0)),color=ycolor,width=3) zAxis = plot(vector((0,0,zSize)),color=zcolor,width=3) #Labels for axes #set fontsize (change if you need to) fsize=20 xT=text3d("X",(xSize*1.1,0,0),color=xcolor,fontsize=fsize) yT=text3d("Y",(0,ySize*1.1,0),color=ycolor,fontsize=fsize) zT=text3d("Z",(0,0,zSize*1.1),color=zcolor,fontsize=fsize) return xAxis+xT+yAxis+yT+zAxis+zT asize = 16 #axis size psize = 12 #plane size #define the normal and distance to origin N=vector((1,1,1)) d=5 #dist to origin #equation of plane is K*<x,y,z>=d ONLY as long as K is a unit normal vector K=N/N.norm() #note syntax for norm of a vector: the norm of A is A.norm() f = K[0]*x+K[1]*y + K[2]*z-d #set this equal to zero to get equation of plane P = plot(axes(asize)) P += implicit_plot3d(f, (x,-psize, psize), (y,-psize, psize), (z, -psize, psize), color='orange', opacity=0.1) P += plot(10*N, width = 3, color='black') #I multiplied N to make it longer. You may want to change this. show('The equation of the plane is $'+latex(f)+'=0$') show('Normal vector is black') show(P, spin =5)
The equation of the plane is 133x+133y+133z5=0 \frac{1}{3} \, \sqrt{3} x + \frac{1}{3} \, \sqrt{3} y + \frac{1}{3} \, \sqrt{3} z - 5 =0
Normal vector is black
3D rendering not yet implemented
a=(2,-1,5) A=vector(a) b=(1,1,1); c=(0,-2,4) B=vector(b);C=vector(c) A.dot_product(B)
66
A.cross_product(B)
(6,3,3)\left(-6,\,3,\,3\right)
B.cross_product(A)
(22,16,1)\left(22,\,-16,\,1\right)
A.cross_product(B)
(22,16,1)\left(-22,\,16,\,-1\right)
A.norm()
29\sqrt{29}
n=2^63 for _ in range(30): p=n.next_prime() print p, p-n n=p
9223372036854775837 29 9223372036854775907 70 9223372036854775931 24 9223372036854775939 8 9223372036854775963 24 9223372036854776063 100 9223372036854776077 14 9223372036854776167 90 9223372036854776243 76 9223372036854776257 14 9223372036854776261 4 9223372036854776293 32 9223372036854776299 6 9223372036854776351 52 9223372036854776393 42 9223372036854776407 14 9223372036854776561 154 9223372036854776657 96 9223372036854776687 30 9223372036854776693 6 9223372036854776711 18 9223372036854776803 92 9223372036854777017 214 9223372036854777059 42 9223372036854777119 60 9223372036854777181 62 9223372036854777211 30 9223372036854777293 82 9223372036854777341 48 9223372036854777343 2
n
9223372036854777343
prod=n*(n-2)
mod(prod,9)
8
prod
85070591730234644163149060928396584963