My question is regarding question 2c in the attached image:
Luke is at the beacon, What direction should Luke head to cool off the fastest?
The answer provided to me was $\langle -2e^4, 4e^4, -2e^4 \rangle$, which is just the gradient of the temperature function evaluated at beacon $(-2, 1, -2)$. Wouldn't going that direction result in the steepest increase of temperature instead of decrease?
Thank you for reading my question!
The gradient is in the direction of steepest ascent. If Luke wants to heat up the fastest, he moves in the direction of the gradient. If he wants to cool off the fastest, he moves opposite the direction of the gradient. The correct solution is in the direction opposite of your solution.