please solve my doubts and explain in detail for questions written in red, answer is provided

4. Classes and function operator overloading: (a) For the MathVector class above, show how the square brackets operato (indexing operator) can be overloaded to return a given indexed element Should this operator be a class member function, and if so will it be private or public? (b) Explain how the * operator can be overloaded to enable expressions such as av, va, and u* v (which we take to mean the cross-product, which you need only implement for 3-vectors), where u and v are vectors and a is a scalar. a 1 class MathVector { public: // constructors and other public interface goes here 2 1 6 7 double koperator [] (int i) { return elements[i]; } private: // private data std::vector elements; int length; 10 11 12 Listing 5: Indexing operator for MathVector celumny posterende bon the time your home easy understanding wory Note that we return a reference to the ith element. This means the indexing operator || behaves exactly as for an array. We could alternatively return just ? a double (not a reference), which would mean we could access the element values, but not set them: mv[i] would be a valid value in a normal expression (e.g. double val = mv[i]*10.0;), but we could not use it in an assignment, viz mv[i] = 10.0;. It should probably) be a class member function as this is more compatible with traditional notation: we don't see the indexing operator as modifying the instance. The overloaded operator must be public otherwise the outside program could not use it. 1 13 MathVector operator+(const double alpha, Mathvector &m) { // pass m by reference, as it might be big! MathVector retval(m); // use the Math Vector copy // constructor: we don't want to // accidentally modify , to which we 1/ have access via the reference. for (int i = 0; i } return retval; 27 9 10 12 13 14 15 16 MathVector operator*(Mathvector &m, const double alpha) { MathVector retval(m); for (int i = 0; i return retval; } 17 18 19 20 Listing 6: Multiplication by scalar please explain 2 Math Vector operator(MathVector &mi, MathVector &m2) { MathVector retval (3); // default MathVector constructor retval [0] = m1[1] + m2 [2] - mi[2] * m1(1): there codes retval [1] - m1[2] + m2(0) - mi[0] + m2 [2] : retval [2] = m1 [0] + m2(1) - m1(1) m2[0]; in detail return retval; } Listing 7: Vector multiplication Note that vector product is defined here only for 3-vectors, so the solution above is a bit of a hack. The declaration Mathvector retval(m); in the first two definitions uses the copy constructor to ensure that the return vector has the same size as the input vector<.size>