% Demonstration of the construction of a Householder reflector
% in three dimensions

 x = [.75;1.25;1];
 vperp = .5*(x+norm(x)*[1;0;0]);
 v = .5*(x-norm(x)*[1;0;0]);
 ax = [-2 2 -2 2 -2 2];     % axis limits for the plots

 figure(1), clf
 plot3(0,0,0,'ko','markersize',8), hold on                     % plot origin
 p1=plot3([0 x(1)], [0 x(2)], [0 x(3)],'k-','linewidth',2);    % plot x
 plot3([x(1)], [x(2)], [x(3)],'k.','markersize',20)
 grid on
 axis(ax)
 legend(p1, 'x')
 title('Householder Reflector Demonstration')
 input('press return')

 p2=plot3([0 norm(x)], [0 0], [0 0],'r-','linewidth',2);       % plot norm(x)*e1
 plot3([norm(x)], [0], [0],'r.','markersize',20)
 axis(ax)
 legend([p1 p2], 'x', '||x|| e_1')
 input('press return')

 p3=plot3([0 vperp(1)], [0 vperp(2)], [0 vperp(3)],'b--','linewidth',2); % plot vperp
 plot3([vperp(1)], [vperp(2)], [vperp(3)],'b.','markersize',20)
 legend([p1 p2 p3], 'x', '||x|| e_1', 'v^\perp')
 axis(ax)
 input('press return')

 p4=plot3([0 v(1)], [0 v(2)], [0 v(3)],'b-','linewidth',2);     % plot v
 plot3([v(1)], [v(2)], [v(3)],'b.','markersize',20) 
 axis(ax)
 legend([p1 p2 p3 p4], 'x', '||x|| e_1', 'v^\perp', 'v')
 input('press return')

% plot the plane orthogonal to v
   [Q,R] = qr([v]);               % construct an orthonormal basis for v-perp
    X = 2*[Q(:,2) Q(:,3) -Q(:,2) -Q(:,3)];
    fill3(X(1,:),X(2,:),X(3,:),[0.7 0.7 1]), hold on