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%md <B> Čelija varijabli </B>

Čelija varijabli

t,x,y,z=var('t,x,y,z')
t,n=var('t,n') a=1/factorial(n)*(cos(n*pi/2))*t^n b=1/factorial(n)*(sin(n*pi/2))*t^n sum(a,n,0,infinity) sum(b,n,0,infinity)
cos(t) sin(t)
t=var('t') r=vector((e^t,t^3+t)) der1=diff(r,t);der1 der2=diff(der1,t);der2 der3=diff(der2,t);der3 r=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2+der3(t=0)*t^3/6 r
(e^t, 3*t^2 + 1) (e^t, 6*t) (e^t, 6) (1/6*t^3 + 1/2*t^2 + t + 1, t^3 + t)
t=var('t') r=vector((e^t,t^3+t)) a1=r a2=r(t=0)+der1(t=0)*t a3=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2 a4=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2+der3(t=0)*t^3/6 parametric_plot(a1,(t,-1.5,1.5),color='green')+parametric_plot(a2,(t,-1,1))+parametric_plot(a3,(t,-3,1),color='red')+parametric_plot(a4,(t,-1,1),color='orange')
t=var('t') r=vector((t^2,t^4,t)) der1=diff(r,t) der2=diff(der1,t) der3=diff(der2,t) der4=diff(der3,t) a1=r a2=r(t=0)+der1(t=0)*t a3=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2 a4=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2+der3(t=0)*t^3/6 a5=r(t=0)+der1(t=0)*t+der2(t=0)*t^2/2+der3(t=0)*t^3/6+der4(t=0)*t^4/24 parametric_plot(a1,(t,-1.5,1.5),color='green')+parametric_plot(a2,(t,-1,1),color='red')+parametric_plot(a3,(t,-3,3),color='red')+parametric_plot(a4,(t,-1,1),color='orange')+parametric_plot(a5,(t,-1,1),color='black')
3D rendering not yet implemented
%md $$\vec{r}(t)=e^t \vec{i}+(t^3+t) \vec{j}$$ $$ \vec{r}(t) \approx (\frac{t^3}{6}+\frac{t^2}{2}+t+1) \vec{\imath}+(t^3+t) \vec{\jmath} $$ $$ \operatorname{funkcija}$$
r(t)=eti+(t3+t)j\vec{r}(t)=e^t \vec{i}+(t^3+t) \vec{j}r(t)(t36+t22+t+1)ı+(t3+t)ȷ\vec{r}(t) \approx (\frac{t^3}{6}+\frac{t^2}{2}+t+1) \vec{\imath}+(t^3+t) \vec{\jmath}funkcija\operatorname{funkcija}

Domacˊa zadacˊa1Domaća \space zadaća \quad 1

Zadatak 01

a=vector([1,2,-4]) b=vector([2,0,3]) c=vector([1,0,1]) show(a.cross_product(b)) show(a.dot_product(b)) show(a.dot_product(b)*c) show(a.dot_product(b.cross_product(c))) show((a.cross_product(b)).dot_product(c)) show((b.cross_product(a)).dot_product(c)) show((a.cross_product(b)).cross_product(c)) show(a.cross_product(b.cross_product(c)))
(6,11,4)\displaystyle \left(6,\,-11,\,-4\right)
10\displaystyle -10
(10,0,10)\displaystyle \left(-10,\,0,\,-10\right)
2\displaystyle 2
2\displaystyle 2
2\displaystyle -2
(11,10,11)\displaystyle \left(-11,\,-10,\,11\right)
(4,0,1)\displaystyle \left(4,\,0,\,1\right)

Zadatak 02

show((2*a+3*b-5*c).cross_product(a-2*b-4*c))
(48,70,34)\displaystyle \left(-48,\,70,\,34\right)
%md Zadatak 03

Zadatak 03

b*(a.dot_product(c))-a*(b.dot_product(c)) ((a.cross_product(b)).cross_product(c))
(-11, -10, 11) (-11, -10, 11)
a.cross_product(b.cross_product(c)) (a.cross_product(b.cross_product(c)))
(4, 0, 1) (4, 0, 1)
%md Zadatak 04

Zadatak 04

2
Vektori su linearno zavisni

Domacˊazadacˊa2.Domaća \quad zadaća \quad 2.

Zadatak 01

t=var('t') a=vector((1,t,t^2)) b=vector((-t,2*t,t^3)) c=vector((t^2,-1,t)) show(factor(a.dot_product(b.cross_product(c))))
(t42t3+3t+2)t2\displaystyle {\left(t^{4} - 2 \, t^{3} + 3 \, t + 2\right)} t^{2}

Zadatak 02.

t=var('t') u=vector((t^3+2*t,sin(t),e^t)) d1=diff(u,t) d2=diff(d1,t) d3=diff(d2,t) show(u,d1,d2,d3)
(t3+2t,sin(t),et)\displaystyle \left(t^{3} + 2 \, t,\,\sin\left(t\right),\,e^{t}\right) (3t2+2,cos(t),et)\displaystyle \left(3 \, t^{2} + 2,\,\cos\left(t\right),\,e^{t}\right) (6t,sin(t),et)\displaystyle \left(6 \, t,\,-\sin\left(t\right),\,e^{t}\right) (6,cos(t),et)\displaystyle \left(6,\,-\cos\left(t\right),\,e^{t}\right)

Zadatak 03

a,t,T=var('a,t,T') u=vector((a*cos(t),a*sin(t))) T=(1+t^2)^0.5 assume(t>0) d1=diff(u,t) d2=diff(T,t) show((d1*1/d2))
(1.00000000000000t2+1asin(t)t,1.00000000000000t2+1acos(t)t)\displaystyle \left(-\frac{1.00000000000000 \, \sqrt{t^{2} + 1} a \sin\left(t\right)}{t},\,\frac{1.00000000000000 \, \sqrt{t^{2} + 1} a \cos\left(t\right)}{t}\right)

Zadatak 05

t,i,j=var('t,i,j') u=((cos(t)*i+sin(t)*j)) factor(taylor(u,t,0,10))
-1/3628800*i*t^10 + 1/362880*j*t^9 + 1/40320*i*t^8 - 1/5040*j*t^7 - 1/720*i*t^6 + 1/120*j*t^5 + 1/24*i*t^4 - 1/6*j*t^3 - 1/2*i*t^2 + j*t + i
t=var('t') y(t)=t x(t)=1 parametric_plot( (1,t), (t,-5,5),thickness=1 ,color='cyan',aspect_ratio=true)+parametric_plot( (1-(t^2)/2,t), (t,-3,3),thickness=1 ,color='green',aspect_ratio=true)+parametric_plot( (1-(t^2)/2,t-(t^3)/6), (t,-3,3),thickness=1 ,color='orange',aspect_ratio=true)+parametric_plot( (cos(t),sin(t)),(t,0,2*pi),thickness=1,color='red',aspect_ratio=true)+point((1,0),size=30,color='black')+parametric_plot( (1-(t^2)/2+(t^4)/24-(t^6)/720+(t^8)/40320,t-(t^3)/6+(t^5)/120+(t^7)/5040+(t^9)/362880), (t,-4,4),thickness=1 ,color='magenta',aspect_ratio=true)

Zadatak 06

t,i,j=var('t,i,j') u=((i*e^t+(t^3+t)*j)) factor(taylor(u,t,0,10))
1/3628800*i*t^10 + 1/362880*i*t^9 + 1/40320*i*t^8 + 1/5040*i*t^7 + 1/720*i*t^6 + 1/120*i*t^5 + 1/24*i*t^4 + 1/6*i*t^3 + j*t^3 + 1/2*i*t^2 + i*t + j*t + i

Domacˊazadacˊa3.Domaća \quad zadaća \quad 3.

parametric_plot3d((2*sin(t),2*cos(t),t/2),(t,0,5*pi))
3D rendering not yet implemented
parametric_plot3d(((2*sin(t))^2,4*sin(t)*cos(t),2*cos(t)),(t,0,2*pi))
3D rendering not yet implemented
parametric_plot3d(((cos(t)*e^t,sin(t)*e^t,2*t)),(t,-pi,pi))
3D rendering not yet implemented
t=var('t') parametric_plot3d((sin(2*t),1-cos(2*t),2*cos(t)),(t,0,2*pi))
3D rendering not yet implemented
t,y,z=var('t,y,z') parametric_plot3d((sin(2*t),1-cos(2*t),2*cos(t)),(t,0,2*pi))+implicit_plot3d(x^2+(y-1)^2==1,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='yellow')+implicit_plot3d(x^2+(y)^2+z^2==4,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='green')+implicit_plot3d((y-1)+.5*z^2==1,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='brown')
3D rendering not yet implemented
t,i,j=var('t,i,j') r=((cos(t)*i+(t^2+2*t+1)*j)) factor(taylor(r,t,0,3))
-1/2*i*t^2 + j*t^2 + 2*j*t + i + j
t=var('t') parametric_plot( (cos(t),t^2+2*t+1), (t,-2,2),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot( (1-0.5*t^2,1+2*t+t^2), (t,-3,3),thickness=1 ,color='green',aspect_ratio=true)
t,y,z=var('t,y,z') parametric_plot( (t,t^2,t^3), (t,-2,2),thickness=1 ,color='red',aspect_ratio=true)+implicit_plot3d(x^2-y==0,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='yellow')+implicit_plot3d(x^3-z==0,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='green')
3D rendering not yet implemented
t,y,z=var('t,y,z') parametric_plot( (t,t^2,t^3), (t,-2,2),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot( (t,t^2,0), (t,-2,2),thickness=1 ,color='green',aspect_ratio=true)+parametric_plot( (t,0,t^3), (t,-2,2),thickness=1 ,color='blue',aspect_ratio=true)+parametric_plot( (0,t^2,t^3), (t,-2,2),thickness=1,color='black',aspect_ratio=true)
3D rendering not yet implemented
c=var('c') r=vector((c*t,c*sqrt(2)*ln(t),c/t)) diff(r,t)
(c, sqrt(2)*c/t, -c/t^2)
(simplify(sqrt(c^2+2*c^2/t^2+c^2/t^4)))
sqrt(c^2 + 2*c^2/t^2 + c^2/t^4)
integrate(sqrt(c^2+2*c^2/t^2+c^2/t^4),t,1,10)
99/10*c
parametric_plot( (3*t-3*t^3,3*t^2,3*t+t^3), (t,-1,1),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot( (3*t,0,3*t), (t,-1,1),thickness=1 ,color='green',aspect_ratio=true)+point((0,0,0),size=10)
3D rendering not yet implemented
r=vector((2*t,ln(t),t^2)) d1=diff(r,t) d2=diff(d1,t) d3=diff(d2,t) d1(t=1),d2(t=1),d3(t=1)
((2, 1, 2), (0, -1, 2), (0, 2, 0))
parametric_plot( (2*t,ln(t),t^2), (t,0.1,3),thickness=1 ,color='red',aspect_ratio=true)
3D rendering not yet implemented
t=var('t') a=vector([sin(t)^3,cos(t)^3,cos(t)^2]) b=diff(a,t)(t=pi/4) c=diff(diff(a,t),t)(t=pi/4) b c
(3/4*sqrt(2), -3/4*sqrt(2), -1) (3/4*sqrt(2), 3/4*sqrt(2), 0)
(b.cross_product(c)).cross_product(b)
(39/16*sqrt(2), 39/16*sqrt(2), 0)
implicit_plot3d(z==x^2-y^2,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='yellow')+implicit_plot3d(z==x+y-1,(x,-10,10),(y,-10,10),(z,-10,10),opacity=.40,color='green')
3D rendering not yet implemented
parametric_plot( (sin(2*t)*cos(t),sin(2*t)*sin(t),cos(2*t)), (t,0,2*pi),thickness=1 ,color='green',aspect_ratio=true)+implicit_plot3d(x^2+y^2+z^2==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40,color='brown')
3D rendering not yet implemented

Odredite duljinu luka krivulje r=ti+t2j+t3k \vec{r}=t \vec{i}+t^2 \vec{j}+t^3 \vec{k} .

r=vector((t,t^2,2/3*t^3)) a=diff(r,t) integrate(sqrt(4*(t)^4 + 4*(t)^2 + 1),t,0,2)
22/3
r=vector((t,t^2,2/3*t^3)) a=diff(r,t) a=a.norm() integrate(a,t,0,2)
22/3
r=vector((e^t*cos(t),e^t,e^t*sin(t))) a=diff(r,t) b=diff(a,t) c=diff(b,t) show('$\dot{\\vec{r}}$=',a) show('$\ddot{\\vec{r}}$=',b) show('$\dddot{\\vec{r}}$=',c)
r˙\dot{\vec{r}}= (cos(t)etetsin(t),et,cos(t)et+etsin(t))\displaystyle \left(\cos\left(t\right) e^{t} - e^{t} \sin\left(t\right),\,e^{t},\,\cos\left(t\right) e^{t} + e^{t} \sin\left(t\right)\right)
r¨\ddot{\vec{r}}= (2etsin(t),et,2cos(t)et)\displaystyle \left(-2 \, e^{t} \sin\left(t\right),\,e^{t},\,2 \, \cos\left(t\right) e^{t}\right)
ParseError: KaTeX parse error: Undefined control sequence: \dddot at position 1: \̲d̲d̲d̲o̲t̲{\vec{r}}= (2cos(t)et2etsin(t),et,2cos(t)et2etsin(t))\displaystyle \left(-2 \, \cos\left(t\right) e^{t} - 2 \, e^{t} \sin\left(t\right),\,e^{t},\,2 \, \cos\left(t\right) e^{t} - 2 \, e^{t} \sin\left(t\right)\right)
d=a.cross_product(b) show('$\dot{\\vec{r}}\\times \ddot{\\vec{r}}$=',simplify(d))
r˙×r¨\dot{\vec{r}}\times \ddot{\vec{r}}= (2cos(t)e(2t)(cos(t)et+etsin(t))et,2(cos(t)etetsin(t))cos(t)et2(cos(t)et+etsin(t))etsin(t),(cos(t)etetsin(t))et+2e(2t)sin(t))\displaystyle \left(2 \, \cos\left(t\right) e^{\left(2 \, t\right)} - {\left(\cos\left(t\right) e^{t} + e^{t} \sin\left(t\right)\right)} e^{t},\,-2 \, {\left(\cos\left(t\right) e^{t} - e^{t} \sin\left(t\right)\right)} \cos\left(t\right) e^{t} - 2 \, {\left(\cos\left(t\right) e^{t} + e^{t} \sin\left(t\right)\right)} e^{t} \sin\left(t\right),\,{\left(\cos\left(t\right) e^{t} - e^{t} \sin\left(t\right)\right)} e^{t} + 2 \, e^{\left(2 \, t\right)} \sin\left(t\right)\right)
f=(a.norm())^3 d=(d.norm()) h=d/f show('$\kappa(t)=$',((h(t=5)).n()))
κ(t)=\kappa(t)= 0.00317629867621925\displaystyle 0.00317629867621925
r=vector((10*t,10*sqrt(2)*ln(t),10/t)) a=diff(r,t) a=a.norm() integrate(a,t,1,5)
48
r=vector((e^(2*t)*cos(t),e^(2*t)*sin(t),e^(2*t))) a=diff(r,t);a b=diff(a,t);b d=a.norm() c=a.cross_product(b) f=c.norm() l=f/d^3 (l(t=1/2*ln(1/9*sqrt(5)))).n()
(2*cos(t)*e^(2*t) - e^(2*t)*sin(t), cos(t)*e^(2*t) + 2*e^(2*t)*sin(t), 2*e^(2*t)) (3*cos(t)*e^(2*t) - 4*e^(2*t)*sin(t), 4*cos(t)*e^(2*t) + 3*e^(2*t)*sin(t), 4*e^(2*t)) 1.00000000000000
t,i,j,k=var('t,i,j,k') u=((cos(3*t)*i+(t^2+3*t+1)*j-sin(t)^3*k)) factor(taylor(u,t,0,2))
-9/2*i*t^2 + j*t^2 + 3*j*t + i + j
parametric_plot3d( (t^2+4*t+6,2*t^2+2*t+3,5*t^2+2*t+7), (t,-5,5),thickness=1 ,color='red',aspect_ratio=true)+implicit_plot3d(x-3*y+z-4==0,(x,-100,150),(y,-100,150),(z,-100,150),opacity=.40,color='green')
3D rendering not yet implemented
r=vector((cos(t)+sin(t)^2,sin(t)-sin(t)*cos(t),-cos(t))) show('$\\vec{r}=$',r) a=diff(r,t) b=diff(a,t) c=diff(b,t) d=a(t=pi/2) show('$\\dot{\\vec{r}}=$',a) show('$\\ddot{\\vec{r}}=$',b) show('$t=\\frac{\\pi}{2}$') show('$\\dot{\\vec{r}}(\\frac{\\pi}{2})=$',a(t=pi/2)) d=(d/d.norm()) e=a.cross_product(b)(t=pi/2) f=e/e.norm() show('$\\vec{t}_0=$',d) show('$\\vec{b}_0=$',f) show('$\\vec{n}_0=$',f.cross_product(d))
r=\vec{r}= (sin(t)2+cos(t),cos(t)sin(t)+sin(t),cos(t))\displaystyle \left(\sin\left(t\right)^{2} + \cos\left(t\right),\,-\cos\left(t\right) \sin\left(t\right) + \sin\left(t\right),\,-\cos\left(t\right)\right)
r˙=\dot{\vec{r}}= (2cos(t)sin(t)sin(t),cos(t)2+sin(t)2+cos(t),sin(t))\displaystyle \left(2 \, \cos\left(t\right) \sin\left(t\right) - \sin\left(t\right),\,-\cos\left(t\right)^{2} + \sin\left(t\right)^{2} + \cos\left(t\right),\,\sin\left(t\right)\right)
r¨=\ddot{\vec{r}}= (2cos(t)22sin(t)2cos(t),4cos(t)sin(t)sin(t),cos(t))\displaystyle \left(2 \, \cos\left(t\right)^{2} - 2 \, \sin\left(t\right)^{2} - \cos\left(t\right),\,4 \, \cos\left(t\right) \sin\left(t\right) - \sin\left(t\right),\,\cos\left(t\right)\right)
t=π2t=\frac{\pi}{2}
r˙(π2)=\dot{\vec{r}}(\frac{\pi}{2})= (1,1,1)\displaystyle \left(-1,\,1,\,1\right)
t0=\vec{t}_0= (133,133,133)\displaystyle \left(-\frac{1}{3} \, \sqrt{3},\,\frac{1}{3} \, \sqrt{3},\,\frac{1}{3} \, \sqrt{3}\right)
b0=\vec{b}_0= (11414,1714,31414)\displaystyle \left(\frac{1}{14} \, \sqrt{14},\,-\frac{1}{7} \, \sqrt{14},\,\frac{3}{14} \, \sqrt{14}\right)
n0=\vec{n}_0= (542143,221143,142143)\displaystyle \left(-\frac{5}{42} \, \sqrt{14} \sqrt{3},\,-\frac{2}{21} \, \sqrt{14} \sqrt{3},\,-\frac{1}{42} \, \sqrt{14} \sqrt{3}\right)
parametric_plot3d( (-t^3+3*t,3*t^2,-t^3+3*t), (t,-10,10),thickness=1 ,color='red',aspect_ratio=true)+implicit_plot3d(x-z==0,(x,-970,970),(y,-100,300),(z,-970,970),opacity=.40,color='green')
3D rendering not yet implemented
parametric_plot3d( (t,t^2,t^3), (t,-10,10),thickness=1 ,color='red',aspect_ratio=true)+implicit_plot3d(x==0,(x,-970,970),(y,-100,300),(z,-970,970),opacity=.40,color='green')
3D rendering not yet implemented
x,y,z,t=var('x,y,z,t') implicit_plot3d(x^2+y^2+4*z^2==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)+parametric_plot3d( (sin(t),0,0.5*cos(t)), (t,0,pi),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot3d( (cos(t),sin(t),0), (t,0,2*pi),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot3d( (-sin(t),0,0.5*cos(t)), (t,0,pi),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot3d( (sin(t),0,0.5*cos(t)), (t,0,pi),thickness=1 ,color='red',aspect_ratio=true)+parametric_plot3d( (0.6*sin(t),0.8*sin(t),0.5*cos(t)), (t,0,pi),thickness=1 ,color='red',aspect_ratio=true)
3D rendering not yet implemented
parametric_plot((sin(t),sin(t)),(t,0,2*pi))
implicit_plot3d(x^2/sin(pi/50)+y^2==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)+implicit_plot3d(y^2+4*z^2/(cos(pi/50))==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)
3D rendering not yet implemented
implicit_plot3d(4*z^2-x^2/0.36-1==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40,color='red')+implicit_plot3d(4*z^2-y^2/0.64==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)
3D rendering not yet implemented
implicit_plot3d(x^2+y^2==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)+implicit_plot3d(x+y==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)
3D rendering not yet implemented
%md
u,v=var('u,v') parametric_plot3d(((2+cos(u/2)*sin(v)-sin(u/2)*sin(2*v))*cos(u),(2+cos(u/2)*sin(v)-sin(u/2)*sin(2*v))*sin(u),sin(u/2)*sin(v)+cos(u/2)*sin(2*v)),(u,0,2*pi),(v,0,2*pi),color='green')
3D rendering not yet implemented
x,y,z,t=var('x,y,z,t') implicit_plot3d((x-z)^2+(y-z)^2==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)+implicit_plot3d(z==0,(x,-2,2),(y,-2,2),(z,-2,2),color='green',opacity=.40)+implicit_plot3d(x^2+y^2==1,(x,-2,2),(y,-2,2),(z,-2,2),color='red',opacity=.40)
3D rendering not yet implemented
x,y,z,t=var('x,y,z,t') implicit_plot3d(x^2-y==1,(x,-2,2),(y,-2,2),(z,-2,2),opacity=.40)
3D rendering not yet implemented
t=var('t') parametric_plot3d((e^t,e^(-t),sqrt(2)*t),(t,0,1))
3D rendering not yet implemented
%md Zadatak 2. Kako glase jednadžbe tangencijalnih ravnina plohe $\quad\vec{r}(u,v)=\left \{ u,u+v,u^2+v^2 \right \} \quad$ koje prolaze pravcem $ \quad \frac{x-2}{2}=\frac{y-3}{3}=\frac{z-1}{10}. $

Zadatak 2.

Kako glase jednadžbe tangencijalnih ravnina plohe r(u,v)={u,u+v,u2+v2}\quad\vec{r}(u,v)=\left \{ u,u+v,u^2+v^2 \right \} \quad koje prolaze pravcem x22=y33=z110. \quad \frac{x-2}{2}=\frac{y-3}{3}=\frac{z-1}{10}.

r=vector((u,u+v,u^2+v^2)) ru=diff(r,u) rv=diff(r,v) show('$\\vec{r}(u,v)=$',r) show('$\\vec{r}_{u}=$',ru) show('$\\vec{r}_{v}=$',rv) rurv=ru.cross_product(rv) show('$\\vec{r}_{u}\\times \\vec{r}_{v}=$',rurv) p=vector((4,3,10)) show('$\\vec{p}=$',p) np=rurv.dot_product(p) show('$\\vec{N}\\cdot\\vec{p}=$',np,'$=0$') show('$v=4u-5$') ruu=r(v=4*u-5) show('$\\vec{r}(u)=$',ruu) Nu=rurv(v=4*u-5) show('$\\vec{N}(u)=$',Nu) show('$A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0$') show('$T(2,3,10)$') show('$(6u-10)(2-u)+(-8u+10)(3-5u+5)+1-((4u-5)^2+u^2)=0$') show(solve((6*u-10)*(2-u)+(-8*u+10)*(3-5*u+5)+1-((4*u-5)^2+u^2),u)) show(((6*u-10)*(x-u)+(-8*u+10)*(y-5*u+5)+z-((4*u-5)^2+u^2))(u=2),'$=0$') show(((6*u-10)*(x-u)+(-8*u+10)*(y-5*u+5)+z-((4*u-5)^2+u^2))(u=18/17),'$=0$')
r(u,v)=\vec{r}(u,v)= (u,u+v,u2+v2)\displaystyle \left(u,\,u + v,\,u^{2} + v^{2}\right)
ru=\vec{r}_{u}= (1,1,2u)\displaystyle \left(1,\,1,\,2 \, u\right)
rv=\vec{r}_{v}= (0,1,2v)\displaystyle \left(0,\,1,\,2 \, v\right)
ru×rv=\vec{r}_{u}\times \vec{r}_{v}= (2u+2v,2v,1)\displaystyle \left(-2 \, u + 2 \, v,\,-2 \, v,\,1\right)
p=\vec{p}= (4,3,10)\displaystyle \left(4,\,3,\,10\right)
Np=\vec{N}\cdot\vec{p}= 8u+2v+10\displaystyle -8 \, u + 2 \, v + 10 =0=0
v=4u5v=4u-5
r(u)=\vec{r}(u)= (u,5u5,(4u5)2+u2)\displaystyle \left(u,\,5 \, u - 5,\,{\left(4 \, u - 5\right)}^{2} + u^{2}\right)
N(u)=\vec{N}(u)= (6u10,8u+10,1)\displaystyle \left(6 \, u - 10,\,-8 \, u + 10,\,1\right)
A(xx0)+B(yy0)+C(zz0)=0A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0
T(2,3,10)T(2,3,10)
(6u10)(2u)+(8u+10)(35u+5)+1((4u5)2+u2)=0(6u-10)(2-u)+(-8u+10)(3-5u+5)+1-((4u-5)^2+u^2)=0
[u=(1817)\displaystyle u = \left(\frac{18}{17}\right), u=2\displaystyle u = 2]
2x6y+z+13\displaystyle 2 \, x - 6 \, y + z + 13 =0=0
6217x+2617y+z+2917\displaystyle -\frac{62}{17} \, x + \frac{26}{17} \, y + z + \frac{29}{17} =0=0
u,v=var('u,v') parametric_plot3d((u,u+v,u^2+v^2),(u,-10,10),(v,-10,10),color='green',frame=false)+parametric_plot3d((4*u+2,3*u+3,10*u+1),(u,-10,10),color='red',frame=false)+implicit_plot3d(2*x-6*y+z+13==0,(x,-50,50),(y,-50,50),(z,-50,50),opacity=.40)+implicit_plot3d(-62*x+26*y+17*z+29==0,(x,-50,50),(y,-50,50),(z,-50,50),opacity=.40)
3D rendering not yet implemented