Can anyone solve this? Find the shortest distance between the plane 3*x1+4*x2+x3=1 and the point (-1,1,1). (optimization problem?)

Find the shortest distance between the plane 3*x1+4*x2+x3=1 and the point (-1,1,1)

Answer 1

[Bruno Luong]

n = [3;4;1];
d = abs((n'*[-1;1;1]-1)/sqrt(n'*n))

gives:

d =

    0.1961

[Busra Tabak]

thank you, correct answer but this is not optimization solution. I tried this code

syms x1 x2 x3 r ll

f=3*(x1+1)+4*(x2-1)+(x3-1);

g=3*x1+4*x2+x3-1;

L=f-r*g;

gradL=gradient(L);

[x1s x2s x3s rs]=solve(gradL==0,[x1 x2 x3 r],'Real',true);

h=.01;

k=.01;

for i=1:numel(x1s)

fopt=double(subs(f,[x1 x2 x3],[x1s(i) x2s(i) x3s(i)]));

gc=subs(g,[x1 x2 x3],[x1s(i)+h x2s(i)+k x3s(i)+ll]);

l=double(solve(gc==0,ll));

[a j]=min(abs(l));

l=l(j);

fnear=double(subs(f,[x1 x2 x3],[x1s(i)+h x2s(i)+k x3s(i)+l]));

[x1s(i) x2s(i) x3s(i)]

fopt=sqrt(fopt)

if fopt<fnear

disp('min')

elseif fopt>fnear

disp('max')

end

end

[Bruno Luong]

You are free to prefer 23 lines of code instead of 2.

Answer 2

[Alan Weiss]

If you want to use an optimization technique, see Shortest Distance to a Plane.

Alan Weiss

MATLAB mathematical toolbox documentation