*NEED MATLAB CODES, THANK YOU SO MUCH IN ADVANCE* 1. By using MATLAB find the ar

Question

*NEED MATLAB CODES, THANK YOU SO MUCH IN ADVANCE*

1. By using MATLAB find the area of the triangle with vertices (3,1,0),

(1,1,1) and (0, -2,-1).

Next, we investigate curves in three dimensions. For many of these

problems, the output is a picture. Be sure to title the pictures

using the ``title'' command.

2. Consider the helix (vector)

r(t) = (cos t) i+(sin t) j+t/(2*pi) k for 0<=  t<=  4*pi

where i, j and k are unit vectors. Run the following script.

t=linspace(0,4*pi,101);

x=inline('cos(t)');

y=inline('sin(t)');

z=inline('t/(2*pi)');

plot3(x(t),y(t),z(t))

hold on

% Add the velocity vectors.

for s=linspace(0,4*pi,17);

p=[x(s),y(s),z(s)];

v=[-sin(s),cos(s),1/(2*pi)];

arrow3(p,v,'r')

end

view(135,40)

hold off

3. Repeat problem 2 for the curve r(t) = t i + (t^2/2)j + (t^3/3)k

for -2<=  t<=  2. If you follow

the model of problem 2 beware to write ``y=inline('t.^2/2')" etc.

Now let's calculate lengths of curves. Let C be parametrized by

the vector

r(t)=t i+ (t^2/2) j + (t^3/3)k where 0<= t <= 2.

4. First, we approximate the length of C by making a polygonal

approximation.

t=0:.02:2;

x=t; y=t.^2/2; z=t.^3/3;

sum=0;

for j=1:100

dx=x(j+1)-x(j);

dy=y(j+1)-y(j);

dz=z(j+1)-z(j);

dr=[dx, dy, dz];

sum=sum+norm(dr);

end

disp('Length of the polygonal appox. using 100 seqments')

sum

5. Next we will use the numerical integrator ``quadl''. We have

  ||r'(t)|| = sqrt{1 + t^2 + t^4} .

The arclength integral cannot be computed by hand. So do

speed=inline('sqrt(1+t.^2+t.^4 )')

s=quadl(speed,0,2)

Compare your answer with the answer for problem 4.

6. Let the curve C be parametrized by the vector

P(t) = (1 - cos t)i + (1 + 2t + t^2)j .

(a) Calculate by hand a tangent vector to the curve at P(0).

(b) Use MATLAB to compute the secant vectors (P(t)-P(0))/t for t = .2,

.1, .05. The error

in the secant approximation is

(P(t)-  P(0))/t - P'(0).

By what factor is the error in each component decreased when t is cut in

half?

(c) Plot the curve C for 0<=  t<=  1 and use the {f arrow} feature to

plot each of these secant

vectors as well as the tangent vector computed in part (a). Attach each

of these

vectors to the point P(0) = (0, 1).

Explanation / Answer

To find the area of the triangle formed by the vertices,                  

                                    .

Find :

The area of the triangle is .

Use the MatLab to find the area of the triangle .

MatLab Input & Output:

A = [-2 0 1];

B = [-3 -3 -1];

C = cross(A,B)

C =

   3 -5   6

Now, find the magnitude of the obtained vector by using MatLab.

MatLab Input & Output:

sv=C.*C

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