Cross of the product in any variety of spatial dimensions?

I’m adjusting an ellipse to find out a Kepler orbit for 3 sequential entry factors alongside the orbit route. To illustrate I am utilizing this for an area simulator. An vital factor to remember is that the determinant of the matrix performs a basic function. As I find out about 3×3 matrices, the determinant consists of a cross product operation and a DOT product operation, so it’s geometrically effectively outlined.

Now, I perceive that the cross product is meant to exist just for vectors of three parts and seven parts. Nevertheless, I discovered some code that I altered, from Claude AI, and calculates the determinant for any matrix dimension, offered that the matrix is ​​sq.. Use a so -called cross -product operation for any dimension. What it’s, it really works very effectively for any dimension.

Why does it work, though there is no such thing as a official cross product for dimensions aside from 3 or 7? Is there any title for this so -called cross -product operation that exists in any dimension?

The entire code is in: https://github.com/sjhalayka/ellipse_fitting

// https://claude.ai/chat/3caf4077-28b5-497f-b704-1b0c336a104d
template<size_t N>
class Vector_nD 
{
public:
    std::array<double, N> elements;

    // Helper perform to get the signal of permutation
    static int permutationSign(const std::array<int, (N - 1) > &perm) 
    {
        int inversions = 0;

        for (int i = 0; i < (N - 2); i++) 
            for (int j = i + 1; j < (N - 1); j++) 
                if (perm(i) > perm(j)) inversions++;

        return (inversions % 2 == 0) ? 1 : -1;
    }

public:
    Vector_nD(const std::array<double, N>& comps) : elements(comps) {}

    Vector_nD(void) 
    {
        for (size_t i = 0; i < N; i++)
            elements(i) = 0;
    }


    double operator()(size_t index) const {
        if (index >= N) throw std::out_of_range("Index out of bounds");
        return elements(index);
    }

    static Vector_nD cross_product(const std::vector<Vector_nD<N>>& vectors)
    {
        if (vectors.dimension() != (N - 1))
            throw std::invalid_argument("nD cross product requires precisely (n - 1) vectors");

        std::array<double, N> outcome;// = { 0, 0, 0, 0, 0, 0, 0 };

        for (size_t i = 0; i < N; i++)
            outcome(i) = 0.0;

        // These are the indices we'll use for every part calculation
        std::array<int, (N - 1)> baseIndices;// = { 0, 1, 2, 3, 4, 5 };

        for (int i = 0; i < (N - 1); i++)
            baseIndices(i) = i;

        // For every part of the outcome vector
        for (int okay = 0; okay < N; okay++)
        {
            // Skip okay in our calculations - that is equal to eradicating the k-th column
            // For every permutation of the remaining 6 indices
            do
            {
                // Calculate signal of this time period
                const int signal = permutationSign(baseIndices);

                // Calculate the product for this permutation
                double product = 1.0;

                for (int i = 0; i < (N - 1); i++)
                {
                    const int col = baseIndices(i);
                    // Alter column index if it is previous okay
                    const int actualCol = (col >= okay) ? col + 1 : col;
                    product *= vectors(i)(actualCol);
                }

                outcome(okay) += signal * product;

            } 
            whereas (std::next_permutation(baseIndices.start(), baseIndices.finish()));

            // Reset indices for subsequent part
            for (int i = 0; i < N - 1; i++)
                baseIndices(i) = i;
        }

        return Vector_nD(outcome);
    }

    static double dot_product(const Vector_nD<N>& a, const Vector_nD<N>&b)
    {
        double dot_prod = 0;

        for (size_t i = 0; i < N; i++)
            dot_prod += a(i) * b(i);

        return dot_prod;
    }

    // Methodology to print vector (for debugging)
    void print() const {
        std::cout << "(";
        for (int i = 0; i < N; i++) {
            std::cout << elements(i);
            if (i < (N - 1)) std::cout << ", ";
        }
        std::cout << ")" << std::endl;
    }
};


template <typename size_t N>
double determinant_nxn(const MatrixXd& m)
{
    if (m.cols() != m.rows())
    {
        cout << "Matrix have to be sq." << endl;
        return 0;
    }

    Vector_nD<N> a_nd;
    
    for (size_t i = 0; i < N; i++)
        a_nd.elements(i) = m(0, i);

    std::vector<Vector_nD<N>> vectors;

    for (size_t i = 1; i < N; i++)
    {
        Vector_nD<N> non;

        for (size_t j = 0; j < N; j++)
            non.elements(j) = m(i, j);

        vectors.push_back(non);
    }

    // Compute the cross product utilizing (N - 1) vectors
    Vector_nD<N> outcome = Vector_nD<N>::cross_product(vectors);

    // Flip handedness
    for (size_t i = 0; i < outcome.elements.dimension(); i++)
        if (i % 2 == 1)
            outcome.elements(i) = -result.elements(i);

    double d_dot = Vector_nD<N>::dot_product(a_nd, outcome);
    
    cout << d_dot << endl;
    cout << m.determinant() << endl << endl;

    return d_dot;
}