13.1 Vector-Valued Functions and Space Curves

1-2. Find the domain of the vector functions.


1. .

 Sol) .


2. .

 Sol) and .

     Thus and .


3-5. Find the limit.


3. .

http://matrix.skku.ac.kr/cal-lab/cal-13-1-3.html

 Sol)

  

var('t');

(limit(sin(t)/t,t=0,dir='plus'),limit(cos(t),t=0,dir='plus'),limit(t*ln(t),t=0,dir='plus'));

  (1,0,0)


4. =.


5. .


6-9. Sketch curves with the given vector equations. Indicate with an arrow the direction in which increases.


6. .

http://matrix.skku.ac.kr/cal-lab/cal-13-1-6.html

 Sol)

  

var('t');

x(t)=cos(t);y(t)=sin(t);z(t)=t/2;r(t)=(x(t),y(t),z(t));

dr(t)=(diff(x(t),t),diff(y(t),t,),diff(z(t),t));

t0=2;p1=r(t0);

p2=dr(t0)/abs(dr(t0))+r(t0);

p=parametric_plot3d((x(t),y(t),z(t)),(t,1,5));

utv=line3d([p1,p2],rgbcolor=(1,0,0),arrow_head=True,thickness=3);

show(p+utv);


 

 

7. .

http://matrix.skku.ac.kr/cal-lab/cal-13-1-7.html

 Sol)

  

var('t');

x(t)=t^2;y(t)=t;z(t)=t^3;r(t)=(x(t),y(t),z(t));

dr(t)=(diff(x(t),t),diff(y(t),t,),diff(z(t),t));

t0=2;

p1=r(t0);

p2=dr(t0)/abs(dr(t0))+r(t0);

p=parametric_plot3d((x(t),y(t),z(t)),(t,1,5));

utv=line3d([p1,p2],rgbcolor=(1,0,0),arrow_head=True,thickness=3);

show(p+utv);

 


8. .

http://matrix.skku.ac.kr/cal-lab/cal-13-1-8.html

 Sol)

  

var('t');

x(t)=t;y(t)=t;z(t)=sin(t);r(t)=(x(t),y(t),z(t));

dr(t)=(diff(x(t),t),diff(y(t),t,),diff(z(t),t));

t0=2;

p1=r(t0);

p2=dr(t0)/abs(dr(t0))+r(t0);

p=parametric_plot3d((x(t),y(t),z(t)),(t,1,5));

utv=line3d([p1,p2],rgbcolor=(1,0,0),arrow_head=True,thickness=3);

show(p+utv);


 

 

9. .

http://matrix.skku.ac.kr/cal-lab/cal-13-1-9.html

 Sol)

  

var('t');

x(t)=t^2*cos(t);y(t)=t^2*sin(t);z(t)=sin(t);r(t)=(x(t),y(t),z(t));

dr(t)=(diff(x(t),t),diff(y(t),t,),diff(z(t),t));

t0=2;

p1=r(t0);

p2=dr(t0)/abs(dr(t0))+r(t0);

p=parametric_plot3d((x(t),y(t),z(t)),(t,1,5));

utv=line3d([p1,p2],rgbcolor=(1,0,0),arrow_head=True,thickness=3);

show(p+utv);

 


10-11. Find vector equations and parametric equations for the line segment to .


10. , .

 Sol) ,

      .


11. , .

 Sol) ,

      .


12. Show that the curve with parametric equation , , lies on the cone .

http://matrix.skku.ac.kr/cal-lab/cal-13-1-12.html

 Sol) We have .

  

var('x,y,z,t');

p1=parametric_plot3d((t^2*cos(t),t^2*sin(t),t^2),(t,-pi,pi),color='red');

p2=implicit_plot3d(z^2==x^2+y^2,(x,-10,10),(y,-10,10),(z,0,30),color='yellow',opacity=0.5);

show(p1+p2);


 


  We can see that the curve (red) lies on the cone (yellow).


13. Show that the curve with parametric equations , , is the curve of intersection of the surfaces and .

http://matrix.skku.ac.kr/cal-lab/cal-13-1-13.html

Sol)

  

var('x,y,z,t');

p1=parametric_plot3d((2*sin(t),2*cos(t),t^2),(t,-pi,pi),color='red');

p2=plot3d(x^2/4,(x,-2,2),(y,-2,2),color='green',opacity=0.5);

p3=implicit_plot3d(x^2+y^2==4,(x,-2,2),(y,-2,2),(z,-2,2),color='yellow',opacity=0.5);

show(p1+p2+p3);


 


  We can see that the curve (red) is intersection of two surfaces (yellow, green).


14. Graph the curve with parametric equations

               ,

               ,

               .

http://matrix.skku.ac.kr/cal-lab/cal-13-1-14.html

 Sol)

  

var('t');

parametric_plot3d(((1+sin(4*t))*cos(t),(1+sin(4*t))*sin(t),(1+sin(4*t))),(t,-2*pi,2*pi))


 


15. Show that the curve with parametric equations , , passes through the points and but not through the point .

 Sol) If , , and if , . So, it passes through the points and . But, y-axis of the curve cannot be negative, so it does not pass through .


16-18. Find vector functions that represent the curves of intersection of the two surfaces.


16. The cylinder and the surface .

http://matrix.skku.ac.kr/cal-lab/cal-13-1-16.html

 Sol) Let , . Then .

     So, , is the vector function of intersection of two surfaces.

     Drawing graph

  

var('x,y,z,t');

p1=parametric_plot3d((sqrt(7)*cos(t),sqrt(7)*sin(t),7*sin(2*t)),(t,-pi,pi),color='red');

p2=plot3d(2*x*y,(x,-3,3),(y,-3,3),color='green',opacity=0.5);

p3=implicit_plot3d(x^2+y^2==7,(x,-3,3),(y,-3,3),(z,-3,30),color='yellow',opacity=0.5);

show(p1+p2+p3);


 : See the graph.


17. The planes and .

http://matrix.skku.ac.kr/cal-lab/cal-13-1-17.html

 Sol) Let . Then , .

     Subtracting two equations, , so we have .

     Substituting it to first equation, , so .

     So, is the vector function of intersection of two planes.

 Drawing graph

  

var('x,y,z,t');

p1=parametric_plot3d((t,(t-10)/7,15*(t-3)/14),(t,-3,3),color='red');

p2=implicit_plot3d(3*x-6*y-2*z==15,(x,-3,3),(y,-3,3),(z,-21,6),color='green',opacity=0.5);

p3=implicit_plot3d(2*x+y-2*z==5,(x,-3,3),(y,-3,3),(z,-21,6),color='yellow',opacity=0.5);

show(p1+p2+p3);

  : See the graph.


18. The paraboloid and the parabolic cylinder .

http://matrix.skku.ac.kr/cal-lab/cal-13-1-18.html

 Sol) Put . Then , and .

     So, is the vector function of intersection of two surfaces.

     We can see the answer is correct by drawing the graph of surfaces and vector function.

  

var('x,y,z,t');

p1=parametric_plot3d((t^2,t,t^4+4*t^2),(t,-sqrt(3),sqrt(3)),color='red');

p2=plot3d(x^2+4*y^2,(x,-3,3),(y,-3,3),color='green',opacity=0.5);

p3=implicit_plot3d(x-y^2==0,(x,-3,3),(y,-3,3),(z,-3,45),color='yellow',opacity=0.5);

show(p1+p2+p3);


   


19. If two objects travel through space along two different curves, it is important to know whether they will collide (Will a missile hit its moving target? will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector function.

,

 for .

    Do the particles collide?

 Sol) Two particles collide means has a solution.

     But for second coordinate, hence has no real root, and hence there would be no collision.


 20. Two particles travel along the space curves

, .

          Do the particles collide? Do their paths intersect?

http://matrix.skku.ac.kr/cal-lab/cal-13-1-20.html

 Sol) First coordinate of two particles cannot be same, and hence there would be no collision.

     To see which there is intersection, we draw graph of and .

  

var('t');

p1=parametric_plot3d((t,t^2,t^3),(t,-5,5),color='red');

p2=parametric_plot3d((1+t,2+5*t,3+12*t),(t,-5,5));

show(p1+p2);


 


 By rotating the result graph, we can see that there is one intersection.