MATLAB: Generate a random orthonormal vector (i) to a given unit vector and (ii) with n-2 predetermined components

Question

Hi,

I need to create a random orthonormal vector (i) to a given unit vector and (ii) with (maximal) n-2 predetermined components. For example, say I have the unit vector v1=[-0,52 0,72 -0,19 0,37 -0,18]' (rounded to two digits) and I have the vector v2 with lets say 2 predetermined components that are non-zero, v2=[a b c 0.45 -0.09]. Thus, there should be infinite many solutions for a, b, and c such that v1`*v2=0.

I would like to find an efficient way to generate such a random orthonormal vector v2 where the elements a, b, and c are drawn from a uniform distribution.

I have tried the following:

% start with a random matrix
X = randn(3);
% add some predetermined values
predet=[.4,.5,.4;-.2,-.1,-.2];
V=[X;predet];
% normalize vectors
v1=V(:,1)/norm(V(:,1));
v2=V(:,2)/norm(V(:,2));
% find a random solution for v1'*v2==0 with 3 random entries of v2
check=0;
while check<1
    random = -1 + (1-(-1)).*rand(3,1);
    v2=[random; v2(4:5)];
    % check whether v2 is orthogonal to v1 and v2 has norm 1
    if norm(v2)>0.9999 && norm(v2)<1.0001...
            && v1'*v2<=0.0001 && v1'*v2>=-0.0001
        check=1;
    end
end

However, so far the solution is only a very bad approximation and the algorithm is very slow. Any help on this or suggestions would be very much appreciated.

Answer 1

What about fsolve

b = 1;
c = 1;
v1 = [-0.52 0.72 -0.19 0.37 -0.18];
v2 = @(x)[x b c 0.45 -0.09];
F = @(x) dot(v1,v2(x));
res = fsolve(F,0.5);