Two basic approaches to the generation of conjugate directions are considered for the problem of unconstrained minimization of a quadratic function. Using the principle of choosing a step direction orthogonal to the previous gradient changes, a projected gradient algorithm and a class of variable metric algorithms are derived. Three variants of the class are developed into algorithms, one of which is the Fletcher-Powell-Davidon scheme. Numerical results indicate the merits of the new algorithms compared to several now in use, for a variety of nonquadratic problems.