PKÏRlXñB–Hmimetypetext/x-wxmathmlPKÏRlXQdBV55 format.txt This file contains a wxMaxima session in the .wxmx format. .wxmx files are .xml-based files contained in a .zip container like .odt or .docx files. After changing their name to end in .zip the .xml and eventual bitmap files inside them can be extracted using any .zip file viewer. The reason why part of a .wxmx file still might still seem to make sense in a ordinary text viewer is that the text portion of .wxmx by default isn't compressed: The text is typically small and compressing it would mean that changing a single character would (with a high probability) change big parts of the whole contents of the compressed .zip archive. Even if version control tools like git and svn that remember all changes that were ever made to a file can handle binary files compression would make the changed part of the file bigger and therefore seriously reduce the efficiency of version control wxMaxima can be downloaded from https://github.com/wxMaxima-developers/wxmaxima. It also is part of the windows installer for maxima (https://wxmaxima-developers.github.io/wxmaxima/). If a .wxmx file is broken but the content.xml portion of the file can still be viewed using an text editor just save the xml's text as "content.xml" and try to open it using a recent version of wxMaxima. If it is valid XML (the XML header is intact, all opened tags are closed again, the text is saved with the text encoding "UTF8 without BOM" and the few special characters XML requires this for are properly escaped) chances are high that wxMaxima will be able to recover all code and text from the XML file. PKÏRlXî¨Åm Ï Ï content.xml Kedvenc Matematikai Kísérleteim Alapok Halmazok Számok Valós számok Természetes, egész és racionális számok Megszámlálható halmazok Komplex számok Komplex számok kill(all); (%o0) done A komplex számok definiálásához műveleteket vezetünk be a valós számpárok között. infix("c+"); "c+"(z,w):=[z[1]+w[1],z[2]+w[2]]; (%o1) c+(%o2) z c+ w:=[z1+w1,z2+w2] [1,2] c+ [7,11]; (%o3) [8,13] infix("c*"); "c*"(z,w):=[z[1]*w[1]-z[2]*w[2],z[1]*w[2]+z[2]*w[1]]; (%o4) c*(%o5) z c* w:=[z1*w1−z2*w2,z1*w2+z2*w1] [1,2] c* [7,11]; (%o6) [−15,25] [x,y] c+ [0,0]; (%o7) [x,y] [x,y] c+ [-x,-y]; (%o8) [0,0] [x,y] c* [x/(x^2+y^2),-y/(x^2+y^2)]; (%o9) [y2y2+x2+x2y2+x2,0] ratsimp(%); (%o10) [1,0] [x,0] c+ [y,0]; (%o11) [y+x,0] [x,0] c* [y,0]; (%o12) [x*y,0] i:[0,1]; (%o13) [0,1] i c* i; (%o14) [−1,0] Mivel a Maxima eleve komplex számokkal számol, nincs szükség erre. Az itt i-vel jelölt komplex szám jele %i. kill(all); (%o0) done z:x+y*%i; realpart(z); imagpart(z); (%o1) %i*y+x(%o2) x(%o3) y conjugate(z); conjugate(%); conjugate(1/z); (%o4) x−%i*y(%o5) %i*y+x(%o6) 1x−%i*y Példa z:64/(sqrt(3)+%i); (%o7) 64%i+3 rectform(z); realpart(z); imagpart(z); conjugate(z); rectform(%); (%o12) 16*3−16*%i(%o13) 16*3(%o14) −16(%o15) 643−%i(%o16) 16*%i+16*3 Komplex számok abszolút értéke abs(z); cabs(z); (%o17) 32(%o18) 32 z:x+y*%i; abs(z); cabs(z); conjugate(z)/cabs(z)^2; %*z; (%o19) %i*y+x(%o20) %i*y+x(%o21) y2+x2(%o22) x−%i*yy2+x2(%o23)

x−%i*y

*

%i*y+x

y2+x2
expand(%); (%o24) y2y2+x2+x2y2+x2 ratsimp(%); (%o25) 1 Feladat Feladat Feladat Feladat (1) 2/(1-%i)/(3+%i); rectform(%); (%o26) 2

1−%i

*

%i+3

(%o27) %i5+25
(2) 1/(3+4*%i)^2; rectform(%); (%o28) 1

4*%i+3

2
(%o29) −24*%i625−7625
(3) (2+%i)/%i/(4*%i-3); rectform(%); (%o30) −%i*

%i+2

4*%i−3
(%o31) 2*%i25−1125
(4) (sqrt(3)+%i)/(1-%i)/(sqrt(3)-%i); rectform(%); (%o32) %i+3

1−%i

*

3−%i

(%o33)

34+14

*%i−34+14
(5) 1/%i/(3-2*%i)/(1+%i); rectform(%); (%o34) −%i

3−2*%i

*

%i+1

(%o35) −5*%i26−126
(6) %i/(1-%i)/(1-2*%i)/(1+2*%i); rectform(%); (%o36) %i

1−2*%i

*

1−%i

*

2*%i+1

(%o37) %i10−110
Feladat z:1-5*%i; w:3+4*%i; (%o38) 1−5*%i(%o39) 4*%i+3 (1) z/w; rectform(%); (%o40) 1−5*%i4*%i+3(%o41) −19*%i25−1725 (2) conjugate(z)/w; rectform(%); (%o42) 5*%i+14*%i+3(%o43) 11*%i25+2325 (3) z/conjugate(w); rectform(%); (%o44) 1−5*%i3−4*%i(%o45) 2325−11*%i25 (4) conjugate(z/w); rectform(%); (%o46) 5*%i+13−4*%i(%o47) 19*%i25−1725 (5) z/cabs(w); rectform(%); (%o48) 1−5*%i5(%o49) 15−%i (6) cabs(z/w); rectform(%); (%o50) 265(%o51) 265 Feladat z:1+%i; w:1-2*%i; (%o52) %i+1(%o53) 1−2*%i (1) z-z/w; rectform(%); (%o54) −%i+11−2*%i+%i+1(%o55) 2*%i5+65 (2) (z-1)/w; rectform(%); (%o56) %i1−2*%i(%o57) %i5−25 (3) z^2-%i*z/w; rectform(%); (%o58)

%i+1

2
−%i*

%i+1

1−2*%i
(%o59) 11*%i5+35
(4) z/(%i*w); rectform(%); (%o60) −%i*

%i+1

1−2*%i
(%o61) %i5+35
(5) z/cabs(w); rectform(%); (%o62) %i+15(%o63) %i5+15 Feladat z1:2+%i; z2:3-2*%i; z3:-1/2+sqrt(3)*%i/2; (%o64) %i+2(%o65) 3−2*%i(%o66) 3*%i2−12 (1) cabs(3*z1-4*z2)+z3*conjugate(z3); rectform(%); (%o67)

−3*%i2−12

*

3*%i2−12

+157(%o68) 157+1
(2) z1^3+3*z1^2+4*z1-8; rectform(%); (%o69)

%i+2

3
+3*

%i+2

2
+4*

%i+2

−8(%o70) 27*%i+11
(3) cabs((2*z2+z1-5-%i)/(2*z1-z2+3-%i)); rectform(%); (%o71) 1(%o72) 1 Komplex számok argumentuma és trigonometrikus alakja carg(1+%i); bfloat(%); (%o73) %pi4(%o74) 7.853981633974483b−1 polarform(1+%i); (%o75) 2*%e%i*%pi4 Később tanulandó összefüggés szerint %e^(%i*φ)=cosφ+%i*sinφ. demoivre(%); (%o76) 2*

%i2+12

ratsimp(%); (%o77) %i+1 Példa z:16*sqrt(3)-16*%i; carg(z); (%o78) 16*3−16*%i(%o79) −%pi6 polarform(z); (%o80) 32*%e−%i*%pi6 Gyökvonás komplex számból kill(all); (%o0) done z:R*%e^(%i*φ); r:R^(1/n); w:r*%e^(%i*(φ/n+2*k*%pi/n)); %^n; (%o1) R*%e%i*φ(%o2) R1n(%o3) R1n*%e%i*

φn+2*%pi*kn

(%o4)

R1n*%e%i*

φn+2*%pi*kn

n
radcan(%); (%o5) R*%e%i*φ+2*%i*%pi*k Példa w:2*%e^(-%pi/30+2*k*%pi/5); (%o6) 2*%e2*%pi*k5−%pi30 w^5; (%o7) 32*%e5*

2*%pi*k5−%pi30

expand(%); (%o8) 32*%e2*%pi*k−%pi6 Bővített komplex számok kill(all); (%o0) done infinity; inf; minf; abs(infinity); (%o1) infinity(%o2) inf(%o3) −inf(%o4) inf Feladat (1) polarform(%i); (%o5) %e%i*%pi2 (2) polarform(conjugate(%i)); (%o6) %e−%i*%pi2 (3) polarform(1+%i); (%o7) 2*%e%i*%pi4 (4) polarform(1+%i*sqrt(3)); (%o8) 2*%e%i*%pi3 (5) polarform(sqrt(3)-%i); (%o9) 2*%e−%i*%pi6 Feladat sqrt(%i); polarform(%); (%o10)

−1

14
(%o11) %e%i*%pi4
Feladat Feladat kill(all); (%o0) done (1) sqrt(%i); polarform(%); (%o1)

−1

14
(%o2) %e%i*%pi4
(2) sqrt(conjugate(%i)); polarform(%); (%o3) −%i(%o4) %e−%i*%pi4 (3) sqrt(3+4*%i); polarform(%); (%o5) 4*%i+3(%o6) 5*%e%i*atan

43

2
(4) sqrt(-7+24*%i); polarform(%); (%o7) 24*%i−7(%o8) 5*%e%i*

%pi−atan

247

2
Feladat Feladat kill(all); (%o0) done (1) ((1+2*%i)^2-(1-%i)^3)/ ((3+2*%i)^3-(2+%i)^2); (%o1)

2*%i+1

2
−

1−%i

3

2*%i+3

3
−

%i+2

2
rectform(%); (%o2) 22159−5*%i318 (2) ((1-%i)^5-1)/((1+%i)^5+1); (%o3)

1−%i

5
−1

%i+1

5
+1
rectform(%); (%o4) −32*%i25−125 (3) 2; (1+%i)^16/2^7; (%o5) 2(%o6)

%i+1

16
128
rectform(%); (%o7) 2 (4) -2^26; (1+%i)^52; (%o8) −67108864(%o9)

%i+1

52
rectform(%); (%o10) −67108864 (5) 2^18*(1+%i*sqrt(3))^2/-2^10; (%o11) −256*

3*%i+1

2
rectform(%); (%o12) 512−512*3*%i ((1+%i*sqrt(3))/(1-%i))^20; (%o13)

3*%i+1

20

1−%i

20
rectform(%); (%o14) 512−512*3*%i Feladat kill(all); (%o0) done declare(a,real); declare(b,real); declare(c,real); ω:-1/2+%i*sqrt(3)/2; (%o1) done(%o2) done(%o3) done(%o4) 3*%i2−12 (1) (a*ω+b*ω^2)*(a*ω^2+b*ω); (%o5)

3*%i2−12

*b+

3*%i2−12

2
*a

*

3*%i2−12

2*b+

3*%i2−12

*a

ratsimp(%); (%o6) b2−a*b+a2 (2) (a+b+c)*(a+b*ω+c*ω^2)*(a+b*ω^2+c*ω); (%o7)

c+b+a

*

3*%i2−12

*c+

3*%i2−12

2
*b+a

*

3*%i2−12

2*c+

3*%i2−12

*b+a

ratsimp(%); (%o8) c3−3*a*b*c+b3+a3 (3) (a+b*ω+c*ω^2)^3+(a+b*ω^2+c*ω)^3; (%o9)

3*%i2−12

2
*c+

3*%i2−12

*b+a

3+

3*%i2−12

*c+

3*%i2−12

2
*b+a

3
ratsimp(%); (%o10) 2*c3+

−3*b−3*a

*c2+

−3*b2+12*a*b−3*a2

*c+2*b3−3*a*b2−3*a2*b+2*a3
Feladat Feladat kill(all); (%o0) done (1) r:cos(2*t)^(1/2); (%o1) cos

2*t

plot2d([parametric,r*cos(t),r*sin(t), [t,-4,20]],same_xy)$ plot2d: expression evaluates to non-numeric value somewhere in plotting range. (2) r:sin(2*t)^(1/2); (%o3) sin

2*t

plot2d([parametric,r*cos(t),r*sin(t), [t,0,20]],same_xy)$ plot2d: expression evaluates to non−numeric value somewhere in plotting range. (3) (r^2-6*r*cos(t))^2=r^2; (%o5)

sin

2*t

−6*cos

t

*sin

2*t

2
=sin

2*t

r:1+6*cos(t); (%o6) 6*cos

t

+1
plot2d([parametric,r*cos(t),r*sin(t), [t,-4,4]],same_xy)$ r:-1+6*cos(t); (%o8) 6*cos

t

−1
plot2d([parametric,r*cos(t),r*sin(t), [t,-4,4]],same_xy)$ (4) r:4+4*cos(t); (%o10) 4*cos

t

+4
plot2d([parametric,r*cos(t),r*sin(t), [t,-4,4]],same_xy)$ r:-4+4*cos(t); (%o12) 4*cos

t

−4
plot2d([parametric,r*cos(t),r*sin(t), [t,-4,4]],same_xy)$ (5) r:8*sin(t)*cos(t)^2/(sin(t)^4+cos(t)^4); (%o14) 8*cos

t

2
*sin

t

sin

t

4
+cos

t

4
plot2d([parametric,r*cos(t),r*sin(t), [t,-4,4]],same_xy)$ (6) r:1/sqrt(cos(t)^4+sin(t)^4); (%o16) 1sin

t

4
+cos

t

4
plot2d([parametric,r*cos(t),r*sin(t), [t,-4,4]],same_xy)$ (7) r:sqrt(cos(t)^2*sin(t)^2); (%o18) cos

t

*sin

t

plot2d([parametric,r*cos(t),r*sin(t), [t,-4,4]],same_xy)$ (8) r:sqrt(cos(t)*sin(t)); (%o20) cos

t

*sin

t

plot2d([parametric,r*cos(t),r*sin(t), [t,-4,4]],same_xy)$ plot2d: expression evaluates to non−numeric value somewhere in plotting range. Feladat kill(all); (%o0) done (1) r:t; (%o1) t plot2d([parametric,r*cos(t),r*sin(t), [t,0,20]],same_xy)$ (2) r:%pi/t; (%o3) %pit plot2d([parametric,r*cos(t),r*sin(t), [t,1,10]],same_xy)$ (3) r:t/(1+t); (%o5) tt+1 plot2d([parametric,r*cos(t),r*sin(t), [t,0,20]],same_xy)$ (4) r:2^(t/2/%pi); (%o7) 2t2*%pi plot2d([parametric,r*cos(t),r*sin(t), [t,0,20]],same_xy)$ (5) r:2*(1+cos(t)); (%o9) 2*

cos

t

+1

plot2d([parametric,r*cos(t),r*sin(t), [t,0,20]],same_xy)$ (6) r:sin(3*t); (%o11) sin

3*t

plot2d([parametric,r*cos(t),r*sin(t), [t,0,20]],same_xy)$ (7) r:cos(2*t)^(1/2); (%o13) cos

2*t

plot2d([parametric,r*cos(t),r*sin(t), [t,0,20]],same_xy)$ plot2d: expression evaluates to non−numeric value somewhere in plotting range. Feladat Feladat Feladat Feladat Feladat Feladat Feladat Feladat Kvaterniók kill(all); (%o0) done infix("q+"); "q+"(p,q):=[p[1]+q[1],p[2]+q[2]]; (%o1) q+(%o2) p q+ q:=[p1+q1,p2+q2] infix("q*"); "q*"(p,q):=[p[1]*q[1]-conjugate(q[2])*p[2],q[2]*p[1]+p[2]*conjugate(q[1])]; (%o3) q*(%o4) p q* q:=[p1*q1−q2*p2,q2*p1+p2*q1] declare(z,complex); declare(w,complex); (%o5) done(%o6) done [z,w] q+ [0,0]; [z,w] q+ [-z,-w]; (%o7) [z,w](%o8) [0,0] [z,w] q* [1,0]; r:abs(z)^2+abs(w)^2; [z,w] q* [conjugate(z)/r,-w/r]; ratsimp(%); (%o9) [z,w](%o10) z2+w2(%o11) [z*zz2+w2+w*wz2+w2,0](%o12) [z*z+w*wz2+w2,0] [z,0] q+ [w,0]; [z,0] q* [w,0]; (%o13) [z+w,0](%o14) [w*z,0] j:[0,1]; j q* j; [z,0] q+ ([w,0] q* j); (%o15) [0,1](%o16) [−1,0](%o17) [z,w] k:[0,%i]; k q* k; i:[%i,0]; i q* i; (%o18) [0,%i](%o19) [−1,0](%o20) [%i,0](%o21) [−1,0] declare(a,real); declare(b,real); declare(c,real); declare(d,real); (%o22) done(%o23) done(%o24) done(%o25) done [a,0] q* [z,w]; [z,w] q* [a,0]; (%o26) [a*z,a*w](%o27) [a*z,a*w] j q* [z,0]; [z,0] q* j; (%o28) [0,z](%o29) [0,z] realpartq(p):=realpart(p[1]); (%o30) realpartq

p

:=#{Lisp function}

p1

p:[a+%i*b,c+%i*d]; realpartq(p); (%o31) [%i*b+a,%i*d+c](%o32) a q:[z,w]; realpartq(q); (%o33) [z,w](%o34) realpart

z

vectpartq(p):=[imagpart(p[1]), realpart(p[2]),imagpart(p[2])]; (%o35) vectpartq

p

:=[#{Lisp function}

p1

,#{Lisp function}

p2

,#{Lisp function}

p2

]
vectpartq(p); vectpartq(q); (%o36) [b,c,d](%o37) [imagpart

z

,realpart

w

,imagpart

w

]
tv2q(t,v):=[t+%i*v[1],v[2]+%i*v[3]]; (%o38) tv2q

t,v

:=[t+%i*v1,v2+%i*v3]
p:tv2q(a,[b,c,d]); (%o39) [%i*b+a,%i*d+c] conjugateq(p):=tv2q(realpartq(p), -vectpartq(p)); conjugateq(p); (%o40) conjugateq

p

:=tv2q

realpartq

p

,−vectpartq

p

(%o41) [a−%i*b,−%i*d−c]
Kvaterniók abszolút értéke qabs(p):=sqrt(abs(p[1])^2+abs(p[2])^2); (%o42) qabs

p

:=p12+p22
p q* conjugateq(p); (%o43) [

a−%i*b

*

%i*b+a

−

%i*d−c

*

%i*d+c

,

%i*b+a

*

%i*d+c

+

%i*b+a

*

−%i*d−c

]
expand(%); (%o44) [d2+c2+b2+a2,0] p q* conjugateq(p)/qabs(p)^2; (%o45) [

a−%i*b

*

%i*b+a

−

%i*d−c

*

%i*d+c

%i*d+c2+%i*b+a2
,

%i*b+a

*

%i*d+c

+

%i*b+a

*

−%i*d−c

%i*d+c2+%i*b+a2
]
ratsimp(%); (%o46) [d2+c2+b2+a2%i*d+c2+%i*b+a2,0] Feladat Feladat Feladat Feladat Vektoriális szorzás infix("qx*"); "qx*"(p,q):=((p q* q) q+ (-q q* p))/2; (%o47) qx*(%o48) p qx* q:=p q* q q+

−q q* p

2
declare(a1,real); declare(b1,real); declare(c1,real); declare(d1,real); (%o49) done(%o50) done(%o51) done(%o52) done q:[a1+%i*b1,c1+%i*d1]; (%o53) [%i*b1+a1,%i*d1+c1] p qx* q; (%o54) [

c−%i*d

*

%i*d1+c1

−

%i*d+c

*

c1−%i*d1

2
,

%i*b+a

*

%i*d1+c1

−

a−%i*b

*

%i*d1+c1

−

%i*b1+a1

*

%i*d+c

+

a1−%i*b1

*

%i*d+c

2
]
expand(%); (%o55) [%i*c*d1−%i*c1*d,−b*d1+b1*d+%i*b*c1−%i*b1*c] q qx* p; (%o56) [

%i*d+c

*

c1−%i*d1

−

c−%i*d

*

%i*d1+c1

2
,−

%i*b+a

*

%i*d1+c1

+

a−%i*b

*

%i*d1+c1

+

%i*b1+a1

*

%i*d+c

−

a1−%i*b1

*

%i*d+c

2
]
expand(%); (%o57) [%i*c1*d−%i*c*d1,b*d1−b1*d−%i*b*c1+%i*b1*c] i qx* j; j qx* k; k qx* i; j qx* i; k qx* j; i qx* k; (%o58) [0,%i](%o59) [%i,0](%o60) [0,1](%o61) [0,−%i](%o62) [−%i,0](%o63) [0,−1] declare(a2,real); declare(b2,real); declare(c2,real); declare(d2,real); (%o64) done(%o65) done(%o66) done(%o67) done r:[a2+%i*b2,c2+%i*d2]; p:[a+%i*b,c+%i*d]; (%o68) [%i*b2+a2,%i*d2+c2](%o69) [%i*b+a,%i*d+c] ((p qx* q) qx* r) q+ ((q qx* r) qx* p) q+ ((r qx* p) qx* q); (%o70) [

%i*d+c

*

−

%i*b1+a1

*

c2−%i*d2

+

a1−%i*b1

*

c2−%i*d2

+

%i*b2+a2

*

c1−%i*d1

−

a2−%i*b2

*

c1−%i*d1

2
−

c−%i*d

*

%i*b1+a1

*

%i*d2+c2

−

a1−%i*b1

*

%i*d2+c2

−

%i*b2+a2

*

%i*d1+c1

+

a2−%i*b2

*

%i*d1+c1

2
2+

%i*d1+c1

*

%i*b+a

*

c2−%i*d2

−

a−%i*b

*

c2−%i*d2

−

%i*b2+a2

*

c−%i*d

+

a2−%i*b2

*

c−%i*d

2
−

c1−%i*d1

*

−

%i*b+a

*

%i*d2+c2

+

a−%i*b

*

%i*d2+c2

+

%i*b2+a2

*

%i*d+c

−

a2−%i*b2

*

%i*d+c

2
2+

−

%i*b+a

*

c1−%i*d1

+

a−%i*b

*

c1−%i*d1

+

%i*b1+a1

*

c−%i*d

−

a1−%i*b1

*

c−%i*d

*

%i*d2+c2

2
−

%i*b+a

*

%i*d1+c1

−

a−%i*b

*

%i*d1+c1

−

%i*b1+a1

*

%i*d+c

+

a1−%i*b1

*

%i*d+c

*

c2−%i*d2

2
2,

%i*d+c

*

c1−%i*d1

*

%i*d2+c2

−

%i*d1+c1

*

c2−%i*d2

2
−

%i*d+c

*

%i*d1+c1

*

c2−%i*d2

−

c1−%i*d1

*

%i*d2+c2

2
−

%i*b+a

*

%i*b1+a1

*

%i*d2+c2

−

a1−%i*b1

*

%i*d2+c2

−

%i*b2+a2

*

%i*d1+c1

+

a2−%i*b2

*

%i*d1+c1

2
+

a−%i*b

*

%i*b1+a1

*

%i*d2+c2

−

a1−%i*b1

*

%i*d2+c2

−

%i*b2+a2

*

%i*d1+c1

+

a2−%i*b2

*

%i*d1+c1

2
2+−

%i*d1+c1

*

c−%i*d

*

%i*d2+c2

−

%i*d+c

*

c2−%i*d2

2
+

%i*d1+c1

*

%i*d+c

*

c2−%i*d2

−

c−%i*d

*

%i*d2+c2

2
−

%i*b1+a1

*

−

%i*b+a

*

%i*d2+c2

+

a−%i*b

*

%i*d2+c2

+

%i*b2+a2

*

%i*d+c

−

a2−%i*b2

*

%i*d+c

2
+

a1−%i*b1

*

−

%i*b+a

*

%i*d2+c2

+

a−%i*b

*

%i*d2+c2

+

%i*b2+a2

*

%i*d+c

−

a2−%i*b2

*

%i*d+c

2
2+

c−%i*d

*

%i*d1+c1

−

%i*d+c

*

c1−%i*d1

*

%i*d2+c2

2
−

%i*d+c

*

c1−%i*d1

−

c−%i*d

*

%i*d1+c1

*

%i*d2+c2

2
−

%i*b2+a2

*

%i*b+a

*

%i*d1+c1

−

a−%i*b

*

%i*d1+c1

−

%i*b1+a1

*

%i*d+c

+

a1−%i*b1

*

%i*d+c

2
+

a2−%i*b2

*

%i*b+a

*

%i*d1+c1

−

a−%i*b

*

%i*d1+c1

−

%i*b1+a1

*

%i*d+c

+

a1−%i*b1

*

%i*d+c

2
2]
expand(%); (%o71) [0,0] Kvaterniók és a három dimenziós euklidészi tér v:[b,c,d]; v1:[b1,c1,d1]; p:tv2q(0,v); p1:tv2q(0,v1); (%o72) [b,c,d](%o73) [b1,c1,d1](%o74) [%i*b,%i*d+c](%o75) [%i*b1,%i*d1+c1] p2:p q* p1; realpartq(p2); vectpartq(p2); (%o76) [−

%i*d+c

*

c1−%i*d1

−b*b1,%i*b*

%i*d1+c1

−%i*b1*

%i*d+c

](%o77) −d*d1−c*c1−b*b1(%o78) [c*d1−c1*d,b1*d−b*d1,b*c1−b1*c]
infix("s*"); "s*"(v,v1):=-realpartq(tv2q(0,v) q* tv2q(0,v1)); v s* v1; (%o79) s*(%o80) v s* v1:=−realpartq

tv2q

0,v

q* tv2q

0,v1

(%o81) d*d1+c*c1+b*b1
infix("x*"); "x*"(v,v1):=vectpartq(tv2q(0,v) q* tv2q(0,v1)); v x* v1; (%o82) x*(%o83) v x* v1:=vectpartq

tv2q

0,v

q* tv2q

0,v1

(%o84) [c*d1−c1*d,b1*d−b*d1,b*c1−b1*c]
v2:[b2,c2,d2]; v x* (v1 x* v2); (%o85) [b2,c2,d2](%o86) [−d*

b2*d1−b1*d2

−c*

b2*c1−b1*c2

,d*

c1*d2−c2*d1

+b*

b2*c1−b1*c2

,b*

b2*d1−b1*d2

−c*

c1*d2−c2*d1

]
v s* (v1 x* v2); (%o87) b*

c1*d2−c2*d1

+c*

b2*d1−b1*d2

−

b2*c1−b1*c2

*d
A szorzások geometriai jelentése Feladat A belső szorzat vektorokra . a Maxima-ban (1) v1:[3,1,3]; v2:[1,-2,2]; (%o88) [3,1,3](%o89) [1,−2,2] v1 . v2; v1 . v1; v2 . v2; (%o90) 7(%o91) 19(%o92) 9 7/sqrt(9*19); (%o93) 73*19 bfloat(%); (%o94) 5.353033790313108b−1 stb. Feladat kill(all); (%o0) done (1) k; 1; (%o1) k(%o2) 1 (2) 6*k; 36; (%o5) 6*k(%o6) 36 (3) 6*k+k; 49; (%o7) 7*k(%o8) 49 Feladat A vektoriális szorzat ~ a Maxima-ban kill(all); (%o0) done load("vect")$ [1,2,3]~[4,5,6]; (%o2) [1,2,3] ~ [4,5,6] express(%); (%o3) [−3,6,−3] Csak így fejtődik ki. Inkább definiáljuk magunknak. infix("x*"); "x*"(v1,v2):= [v1[2]*v2[3]-v1[3]*v2[2], v1[3]*v2[1]-v1[1]*v2[3], v1[1]*v2[2]-v1[2]*v2[1]]; (%o21) x*(%o22) v1 x* v2:=[v12*v23−v13*v22,v13*v21−v11*v23,v11*v22−v12*v21] Ez is hasznos: uvect(v):=v/(v.v); (%o23) uvect

v

:=vv . v
v:[2,-1,2]; v1:[3,1,5]; v2:[a,2,-1]; (%o7) [2,−1,2](%o8) [3,1,5](%o9) [a,2,−1] v1 x* v2; % . v ; (%o10) [−11,5*a+3,6−a](%o11) −5*a+2*

6−a

−25
ratsimp(%); (%o12) −7*a−13 solve(%=10,a); (%o13) [a=−237] Forgatások Egyéb geometriai alkalmazások Feladat (1) v:[-2,5,1]+t*[-1,2,3]; (%o1) [−t−2,2*t+5,3*t+1] x=v[1]; (%o2) x=−t−2 s:solve(%,t); (%o3) [t=−x−2] s:rhs(s[1]); (%o4) −x−2 subst(s,t,y=v[2]); (%o5) y=2*

−x−2

+5
expand(%); (%o6) y=1−2*x subst(s,t,z=v[3]); (%o7) z=3*

−x−2

+1
expand(%); (%o8) z=−3*x−5 (2) v1:[3,1,2]; v2:[-1,1,3]; (%o9) [3,1,2](%o10) [−1,1,3] v:v1-v2; (%o11) [4,0,−1] v:v1+t*v; (%o12) [4*t+3,1,2−t] y=1; (%o13) y=1 s:solve(z=v[3],t); (%o14) [t=2−z] s:rhs(s[1]); (%o15) 2−z subst(s,t,x=v[1]); (%o16) x=4*

2−z

+3
expand(%); (%o17) x=11−4*z (3) v:[5,1,4+t]; (%o18) [5,1,t+4] x=5; y=1; (%o19) x=5(%o20) y=1 (4) v1:[-2,3,1] x* [2,0,1]; (%o24) [3,4,−6] v:[6,-3,4]+t*%; (%o25) [3*t+6,4*t−3,4−6*t] x=v[1]; (%o26) x=3*t+6 s:solve(%,t); (%o27) [t=x−63] s:rhs(s[1]); (%o28) x−63 subst(s,t,y=v[2]); (%o29) y=4*

x−6

3
−3
expand(%); (%o30) y=4*x3−11 subst(s,t,z=v[3]); (%o31) z=4−2*

x−6

expand(%); (%o32) z=16−2*x Feladat e:[t+2,3*t,2]; (%o33) [t+2,3*t,2] v0:subst(0,t,e); (%o34) [2,0,2] v1:[-2,1,0]; (%o35) [−2,1,0] kill(u,v); (%o38) done s:v0+u*(v1-v0)+v*[1,3,0]; (%o39) [v−4*u+2,3*v+u,2−2*u] Feladat t=x+2; x:t-2; y:t/2; z:-t/2; (%o40) t=x+2(%o41) t−2(%o42) t2(%o43) −t2 s:[3,1,0]+u*[-2,1,-2]+v*[2,1,-1]; (%o44) [2*v−2*u+3,v+u+1,−v−2*u] Feladat n1:[2,-4,2]; n2:[1,-2,1]; (%o45) [2,−4,2](%o46) [1,−2,1] Ha y=z=0, akkor x=1/2, illetve x=1. ([1,0,0] -[1/2,0,0]) . [1,-2,1]/sqrt(6); (%o47) 12*6 Feladat (1) z:2; y:2-t; x:1+3*t; (%o48) 2(%o49) 2−t(%o50) 3*t+1 v:[3,1,0]/sqrt(10); (%o51) [310,110,0] p0:[1,2,2]; (%o52) [1,2,2] ([-2,3,7]-p0) x* v; (%o53) [−510,1510,−610] sqrt((25+225+36)/10); (%o54) 1435 (2) v:[2,-1,3]/sqrt(14); (%o55) [214,−114,314] ([-1,2,1]-[1,12,3]) x* v; (%o56) [−3214,214,2214] sqrt((32^2+2^2+22^2)/14); (%o57) 2*332 (3) z:t-5; y:2*t-1; x:9-4*t; (%o58) t−5(%o59) 2*t−1(%o60) 9−4*t v:[-4,2,1]/sqrt(21); (%o61) [−421,221,121] ([2,2,1]-[9,-1,-5]) x* v; (%o62) [−921,−1721,−221] sqrt((9^2+17^2+2^2)/21); (%o63) 37421 (4) v:[1,-1,3]-[0,2,1]; (%o64) [1,−3,2] v:%/sqrt(14); (%o65) [114,−314,214] ([3,1,2]-[0,2,1]) x* v; (%o66) [114,−514,−814] sqrt((1+25+64)/14); (%o67) 3*57 (5) v:[-1,4,1]/sqrt(18); (%o68) [−13*2,2323,13*2] [-2,4,1] x* v; (%o69) [0,23−13*2,2323−2523] gfactor(%); (%o70) [0,13*2,−2323] sqrt(1/18+8/9); (%o71) 173*2 Feladat (1) y:4-t; x:2*t-4; z:2*t+1; (%o72) 4−t(%o73) 2*t−4(%o74) 2*t+1 [-1,2,2] . [4,-3,-5]/sqrt(9*50); (%o75) −2323 acos(abs(%)); (%o76) acos

2323

float(%), numer; (%o77) 0.3398369094541213 [-1,2,2] x* [4,-3,-5]; (%o78) [−4,3,−5] n:%/sqrt(4^2+3^2+5^2); (%o79) [−2325,35*2,−12] abs(n . ([4,-4,1]-[-5,5,5])); (%o80) 2725+275*2 (2) x:t; y:t; z:t; (%o81) t(%o82) t(%o83) t x:-1/2+5*t; y:3-15*t; z:4*t+6/5; (%o84) 5*t−12(%o85) 3−15*t(%o86) 4*t+65 [1,1,1] . [5,-15,4] /sqrt(3*(5^2+15^2+4^2)); (%o87) −6798 acos(abs(%)); (%o88) acos

6798

float(%), numer; (%o89) 1.356768335994907 n:[1,1,1] x* [5,-15,4]; (%o90) [19,1,−20] n:%/sqrt(19^2+1^2+20^2); (%o91) [19762,1762,−20762] abs(n . [-.5,3,1.2]); (%o92) 30.5762 float(%), numer; (%o93) 1.104898422903383 (3) [3,1,4] . [-1,1,-4]/sqrt(26*18); (%o94) −313 acos(abs(%)); (%o95) acos

313

float(%), numer; (%o96) 0.5880026035475675 n:[3,1,4] x* [-1,1,-4]; (%o97) [−8,8,4] n:n/sqrt(8^2+8^2+4^2); (%o98) [−23,23,13] abs(n . ([5,1,9]-[4,-3,4])); (%o99) 113 Feladat Egyáltalán nincs ilyen sík; ha lenne, normális egységvektora minden koordinátájának abszolút értéke 1/2 lenne, ami lehetetlen. Feladat Tegyük az origót a metszéspontba, és használjunk kvaterniókat.
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