Given a linear objective function $f(\vec{x})=\sum_ia_ix_i$, the direction in which $f$ varies the greatest is known to be $\vec{\nabla}{f}$.
Now given a non-zero vector $\vec{v}$, I am interested in finding the direction $\vec{d}$ in which $f$ varies the greatest, subject to the constraint that $\vec{v}^{T}\vec{d} = 0$. How do I go about finding $\vec{d}$ subject to such an orthogonal constraint?