Question

(1) Draw the Projections of the curve on the three coordinate planes. Use these projections to help sketch the curve. (Hint: a Calculator in parametric mode could help with the first part.)$\vec{r}(t)=< t,sin(t),2cos(t)>$

(2) At what points does the curve $\vec{r}(t)=t\vec{i}+(2t-t^2)\vec{k}$ intersect the paraboloid $z=x^2+y^2$

Answer 1

1) First let us look x-y plane:

It is clear from the expression for the position vector that,

x=t and y=sin(t)

using matlab(code to get every figure is attached at the end) for the ease of doing , the projection on x-y plane will be:

similarily on y-z plane,

projection on x-y plane 1 0.8 0.6 0.4 0.2 y 0 -0.2 -0.4 -0.6 -0.8 1 0 1 2 3 4 5 6 7 х

y=sin(t) and z=2cos(t), therefore projection will be:

projection on y-z plane 2 1.5 1 0.5 N 0 -0.5 -1. -1.5 -2 -1 -0.5 0 0.5 у

Same way,on z-x plane z=2cos(t) and x=t :

Therefore the whole plot in 3D will be:

projection on z-x plane 6 5 4 3 2 1 1 0 -2 -1.5 -1 -0.5 0.5 1 1.5 2 ON

2 1.5 1, in 0 -0.5 -1 -1.5 -2 .5 5 0 0 -0.5 -1 -5

2) It is clear that doesnt have y component. Therefore the intersection is happened at z-x plane. Hence the projection on the x-z plane of these curves reveals the intersection points.

The projection of the second curve on z-x plane has the form .-------------------(1)

Since at intersection,

$7=2$ and (x and component from the first curve )

---------------------(2)

7= 2 -

From (1) and (2),

when x=0 --> z=0 (from (2) or (1)) similarily

when x=1--->z=1

Therefore we uave two intersections, one is at the point (0,0) and second one is at the point(1,1), This is shown in the figure below:

graph to find intersection of the two curves 3 2 1 хо -1 -2 -3 1 1 1 - - 1 -4 -25 -20 -15 -10 -5 0 5 10 15 20 Z

Matlab code to plot figures:

%answer to question(1)
t=0:0.01:2*pi;
x=t;
y=sin(t);
z=2*cos(t);
figure(1)
plot(x,y);
xlabel('x');
ylabel('y');
title('projection on x-y plane');
figure(2)
plot(y,z);
xlabel('y');
ylabel('z');
title('projection on y-z plane');
figure(3)
plot(z,x);
xlabel('z');
ylabel('x');
title('projection on z-x plane');
figure(4);
syms t
xt = t;
yt = sin(t);
zt = 2*cos(t);
fplot3(xt,yt,zt);
%Answer to 2nd question
figure(5)
t1=-4:0.01:4;%new variable for time
x1=t1;
z1=t1.^2;%equivalent to z=x^2
plot(z1,x1);
hold on
z1=2.*t1-t1.^2;
plot(z1,x1);
hold off
xlabel('z');
ylabel('x');
title('graph to find intersection of the two curves');