Given that $g \operatorname { gd } g ( 1,1,5 ) = 3 \vec { i } - 3 \vec { j } + \vec { k }$ , in which of the following directions does g increase the fastest?
A) $7 \vec { i } - 5 \vec { j } + 2 \vec { k }$
B) $\vec { i } - \vec { k }$
C) $- 3 \vec { i } + 3 \vec { j } + 2 \vec { k }$

Question

Quizplus · Accepted Answer

The gradient of $g$, denoted as $\nabla g$, points in the direction of the greatest rate of increase of $g$. Given that $\nabla g(1,1,5) = 3\vec{i} - 3\vec{j} + \vec{k}$, the direction in which $g$ increases the fastest at the point $(1,1,5)$ is parallel to the gradient vector itself. Among the given options, only choice A, $7\vec{i} - 5\vec{j} + 2\vec{k}$, is in the same direction as the gradient vector $3\vec{i} - 3\vec{j} + \vec{k}$ because both vectors can be scaled to match each other, indicating they point in the same direction.

Given That ggd⁡g(1,1,5)=3i⃗−3j⃗+k⃗g \operatorname { gd } g ( 1,1,5 ) = 3 \vec { i } - 3 \vec { j } + \vec { k }ggdg(1,1,5)=3i−3j​+k

Given That $g \operatorname { gd } g ( 1,1,5 ) = 3 \vec { i } - 3 \vec { j } + \vec { k }$