Find the point on a plane which is closest to a given point

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Find the point u on a plane which is closest to a given point b

Define the plane as the span of the vectors [-1;2;1], [1;1;1]. Let b=[0;1;3].

v1 = sym([-1;2;1]); v2 = sym([1;1;1]); b = sym([0;1;3]);
A = [v1,v2];

M = A'*A      % matrix
g = A'*b      % and rhs vector for normal equations

c = M\g       % solve the normal equations
u = A*c

fillpoints([0;0;0],1.6*v1,1.6*(v1+v2),1.6*v2,'c'); hold on
arrow3(v1,'b','LineWidth',2); arrow3(v2,'b','LineWidth',2);
plotpoints(b,u,'ko:'); hold off
alpha(0.6); nice3d; label(b,'b',u,'u')
view(-80,20)

M =
[ 6, 2]
[ 2, 3]
g =
 5
 4
c =
 1/2
   1
u =
 1/2
   2
 3/2

Check that the difference vector is really orthogonal on the plane.

r = u-b        % difference vector
dot(v1,r)      % check that r is orthogonal on v1
dot(v2,r)      % check that r is orthogonal on v2

r =
  1/2
    1
 -3/2
ans =
0
ans =
0

Published with MATLAB® R2014b