Directional Derivatives — Question 4

Question 4 – Solution

Strategy. Decompose the desired unit vector into components parallel and perpendicular to the gradient.

Step 1: Unit basis Let $$e=\frac 1{10}\left\langle 6,8\right\rangle=\left\langle 3/5,4/5\right\rangle,\qquad n=\left\langle-4/5,3/5\right\rangle.$$ These form an orthonormal basis. Write $u=ae+bn$ with $a^2+b^2=1$.

Step 2: Rate condition $$\nabla f\cdot u=10a=5,$$ so $a=1/2$ and $b=\pm\sqrt 3/2$. Thus $$\boxed{u=\frac 12\left\langle 3/5,4/5\right\rangle\pm\frac{\sqrt 3}{2}\left\langle-4/5,3/5\right\rangle}.$$ Equivalently, $$\boxed{u=\left\langle\frac{3\mp 4\sqrt 3}{10},\frac{4\pm 3\sqrt 3}{10}\right\rangle}.$$

Step 3: Verify Orthonormality gives $\|u\|^2=1/4+3/4=1$, and the perpendicular component contributes zero to the dot product, leaving $10(1/2)=5$.