Quiz 4, Exercise #4

Find a point of intersection of the two planes.
Determine the point of intersection from R
Find a vector in the direction of the line of intersection.
Equation of the line
Plot the planes
Plot the line
Set the view and annotate

Find a point of intersection of the two planes.

Set up the augmented matrix for the system, then use Matlab's rref command to place the result in reduced row echelon form.

A=[1 2 -1 4;...
    2, 1, -2 4];
format rat  % rational arithmetic
R=rref(A)

R =
       1              0             -1              4/3     
       0              1              0              4/3

Determine the point of intersection from R

Note that R represents the system

x - z = 4/3

y = 4/3

z = free

If we let z = 0, the x = 4/3, y = 4/3, and z = 0 gives us the point of intersection P(4/3, 4/3, 0).

Find a vector in the direction of the line of intersection.

Take the cross product of vectors normal to each plane.

n1=[1 2 -1];
n2=[2 1 -2];
v=cross(n1,n2)

v =
      -3              0             -3

Equation of the line

Let X=(x,y,z) be an arbitrary point on the line. The PX = t v is the equation of our line. Thus,

< x - 4/3, y - 4/3, z> = t < -3, 0, -3 >

The line has parametric equations

x = 4/3 - 3t

y = 4/3

z = -3t

Plot the planes

[x,y]=meshgrid(0:.2:3);
z1=x+2*y-4;
z2=(2*x+y-4)/2;
phndl1=surf(x,y,z1);
set(phndl1,'EdgeColor','none',...
    'Facecolor',[0.7,0.7,0.7]);
hold on
phndl2=mesh(x,y,z2);
set(phndl2,'EdgeColor','m');

Plot the line

t=linspace(-0.75,0.75);
x=4/3-3*t;
y=4/3*ones(size(t));
z=-3*t;
lhndl=line(x,y,z);
set(lhndl,'Linewidth',2,...
    'Color','b');

Set the view and annotate

xlabel('x-axis')
ylabel('y-axis')
zlabel('z-axis')
title('Intersection of x + 2y -z = 4 and 2x + y - 2z = 4')
view(117,56)