# w = x² - y² + 3z² direction with no change in w

Consider w = x² - y² + 3z². At (1, 1, 1), what is the fastest rate of change for w? What is a direction along which there is no change in w?

I know how to do the first part, since the fastest rate of change is just the value of the gradient at the point. But how do I find a direction along which there is no change in w?

Would the direction be (1, 1, 0)?

Well, the change in the direction $v$ is always the scalar product $$\langle \operatorname{grad} w, v\rangle = \langle (2,-2,6)^t,v\rangle$$ so your answer is correct (hint: there is another direction...)