Question

if the directional derivative of Φ= axy+byz+czx at (1,1,1) has maximum magnitude 4 in a direction parallel to x-axis find value of a,b,c​

Answer 1

A parallel direction (unit vector) to the z-axis is (0,0,1), not (1,2,). Take care not to mix up directions and position vectors.

Explanation-

r(α)=(1,2,α) The position vectors of a line parallel to the z-axis are parameterized, but the direction parallel to the z-axis is a vector pointing down this line. That is, because both the head and tail are on the line, the direction d=(1,2,2)(1,2,1)=(0,0,21). Then, because we only care about the direction and not the magnitude, we simply normalise it to get (0,0,1) or (0,0,1) depending on whether it points up or down.