Generate an orthogonal matrix and a corresponding vector that satisfy certain constraints; see https://math.stackexchange.com/questions/4631555.

Question4631555.java

import java.util.Arrays;
import java.util.Random;

public class Question4631555 {
    final static int l = 4;
    final static int n = (l * (l - 1)) / 2;
    
    final static Random random = new Random();
    
    public static void main(String [] args) {
        double [] [] u = new double [l] [l];
        
        for (int i = 0;i < l;i++)
            randomize (u [i]);

        orthonormalize (u);

        double [] [] m = new double [n] [n];

        for (int i2 = 1,i = 0;i2 < l;i2++)
            for (int i1 = 0;i1 < i2;i1++,i++)
                for (int j2 = 1,j = 0;j2 < l;j2++)
                    for (int j1 = 0;j1 < j2;j1++,j++)
                        m [i] [j] = u [i1] [j1] * u [i2] [j2] - u [i1] [j2] * u [i2] [j1];

        double [] [] p = new double [n] [n];
        
        for (int i = 1;i < n - 1;i++)
            for (int j = 0;j < n;j++)
                p [i] [j] = m [0] [j] * m [i] [j];

        Arrays.fill(p [0],1);
        randomize (p [n - 1]);
        
        orthonormalize (p);
        
        for (int i = 0;i < n;i++)
            p [n - 1] [i]++;
        
        for (int i = 0;i < l;i++) {
            for (int j = 0;j < l;j++)
                System.out.print(u [i] [j] + " ");
            System.out.println();
        }

        System.out.println();
        
        for (int i = 0;i < n;i++)
            System.out.println ("a_" + i + " : " + p [n - 1] [i]);
    }

    static void randomize (double [] x) {
        for (int j = 0;j < x.length;j++)
            x [j] = random.nextDouble();
    }
    
    static double dot (double [] x1,double [] x2) {
        if (x1.length != x2.length)
            throw new IllegalArgumentException();
        double dot = 0;
        for (int k = 0;k < x1.length;k++)
            dot += x1 [k] * x2 [k];
        return dot;
    }
    
    static void orthonormalize (double [] [] x) {
        for (int i = 0;i < x.length;i++) {
            orthogonalize (x,i);
            normalize (x [i]);
        }
    }
    
    static void orthogonalize (double [] [] x,int i) {
        for (int j = 0;j < i;j++)
            orthogonalize (x [i],x [j]);
    }

    static void orthogonalize (double [] x1,double [] x2) {
        double dot = dot (x1,x2);
        for (int k = 0;k < x1.length;k++)
            x1 [k] -= dot * x2 [k];
    }
    
    static void normalize (double [] x) {
        double sum = 0;
        for (int j = 0;j < x.length;j++)
            sum += x [j] * x [j];
        sum = 1 / Math.sqrt(sum);
        for (int j = 0;j < x.length;j++)
            x [j] *= sum;
    }
}