In this problem. You're given these shown information and were asked to first pretend that we are walking in the southern direction due south, in fact, and to determine whether that means we are going up a hill or down a hill. And so to do that, we need first our Grady in Vector for the function. And so we need to take Z with respect to X and that is going to be equal to negative 0.1 x in the partial Ozy, with respect to why it's is going to be equal to again. This was the person of the with respect to X inside the pressure of the with respect to why is going to be equal to negative 0.2 Why now we're going to evaluate these that are given point. Now, we really only need the 60 and 40 in this case because Z is what or Z is a function of X and y So what? We're just gonna use a 60 40. So I'm going to Delhi way at this partial at the 400.60 40 and I get negative 2001 times 60 and that is equal to negative 600.6 in the night of the other way. The negative 0.2 walling at the 0.60 40. And so I got negative 0.2 times 40. And so I get negative 400.8. And so that is our Grady in vector negative 0.6 and negative 0.8. Now we need our direction. Vector. Since they were telling us that we're walking due south, that means that we're not going south or we're going straight down. So which means they're direction Vector is going to be zero in the X direction and negative one in the Y direction. And because we know this and we have to make the length of this doctor one, we're going to divide it by the square root of the some of each of its components squared. So zero squared. Those negative one squared zero squared 01 negative one. Screw this one. So we just get spirit of one, which is one which we are going to divide into each component and just get zero and negative one back. And now we are going to take the dot product of these two vectors, and when we do, we get negative 0.6 common negative 0.8. That product with zero negative one No, 2.6 times zero is zero negative 00.8 times out of one is 10.8. And so our answer is going to be 28 and all I asks us if we're is if we're starting to ascend or descend. And now we know that our label for this is going to be 0.8 meters her every meter we travel forward. Many. We go a 0.8 meters for every meter we go forward and therefore we know that we're ascending or we're going uphill. And so that is the answer to part A and our answer to part B. It asks us about walking in Northwest, and so if we go in Northwest, that's going to be in this direction that is going to have a directional doctor of negative one, because we're going to negative one in the X direction and one in the Y direction because that's how much we're going up. And we've already found our Grady and doctor up here, and so we want to take the DOT product of that with our new vector. Once we make its length of Loch And so I'm going to divide it by the square rule of the solemn each of its components squared. So negative one squared plus one squared negative once credits +11 sport is one. So the spirit of to And so I get negative 1/2 comma, one over route to and again I'm going to typically don't product of the spectre with the new vector So negative 0.6 negative 0.8 and I'm going to write those infractions or negative 0.6 is negative. 3/5 negative 0.8 is negative. 4/5. That product with one over you'd be negative. 1/2 and one over. Route to do the 3/5 sends out of 1/2 is going to be three over five route to and then negative 4/5 times. One or two is going to be my has four over five route too. And so we get negative one over five route to That means we go down 1/5 or two meters for every meter of travel that means again. Then we are in descending. If we walk Northwest's okay and now we want to find the answer Teoh Part C. And to do that, I asks us for the direction in which the slope is the largest. Define that direction. We actually just take our Grady inspector. And so we know our Grady, Inspector. It's right here, and it's also up here you can write in either way. I'm going to choose a torrid as a decimal. And so the direction again, this is the case for all that this happens in all cases for the direction in which a soap is the largest is the grating vector. And so that this direction is going to be negative. 26 negative 260.8. And to find the rate of change in that direction of the grating doctor, we're gonna take the magnitude of it. And so theory of ascent, which is what they want, is going to be equal to the magnitude of the doctor. Negative 0.6, common negative. 28. And when we do that, we got a negative. A 0.6 squared to me the square root of negative 0.6 square, which is 0.36 plus negative pointed squared, which is 0.64 0.36 point 64 is equal to one. We got the spirit of one which is one that's already of a sentence one. And so this is our direction. This is a rate of ascent and now we're asked to find the angle above the horizontal of the path begins. The way we do that is that we take or say that the tangent of the angle that we want is equal to our rate of ascent. So the tangent of some angles equal toe one, which means that the tangent in verse of one is equal to our angle. And we know that a 45 degrees the tangent is equal to one and therefore our angle is equal 2 45 degrees. Again, we got that by taking the tangent of some angle on sending equal to a rate of ascent which we got by taking the magnitude of our direction vector which we got from migrating in factor