[Math] Find a unit vector that minimizes the directional derivative at a point

Question

Find a unit normed vector (direction) $d$ that minimizes the directional derivative, where $\nabla f(x^*)=(12,24)^T$.

My question: Is the directional derivative minimized by selecting a vector orthogonal to the gradient? Is so would $d=(-2,1)^T$

Answer 1

The directional derivative, for every direction defined by a vector $\vec v$, is given by the doct product with the gradient, that is:

$$f_{\vec v}=\frac {\partial f}{\partial \vec v}=\nabla f \cdot \vec v$$

which is

• max (positive) when $\vec v$ is in the direction (i.e. positive multiple) of $\nabla f$
• zero when $\vec v$ is orthogonal to $\nabla f$
• min(negative) when $\vec v$ is in opposite direction with respect to $\nabla f$

thus the unit vector which minimize the directional derivative is

$$u=\left(-\frac1{\sqrt5},-\frac2{\sqrt5} \right)$$