(4A-5)  Here's a contour plot, looking down on our z = x2 +  2y2 surface.

The vectors shown in the centre are in the horizontal plane, z=0. i is a unit vector (length = 1) in the x-direction and j is a unit vector in the y-direction. 

Point A is at (-0.5, -0.5) which, as you recall, is the point where we have been measuring our gradients. Radiating from this point in the horizontal z=0.75 plane are two unit vectors: we've got them pointing backwards (ie -i and -j) just because this makes it easier to visualise what's going on, in this case. 

Our starting point A is in the plane x=y and if we move by equal amounts in the horizontal x and y-directions, we will reach another point in this plane. So the resultant of adding these two vectors, is a vector that points in our desired direction, shown in the picture as vector v.

①  What is the length of vector v?

Now lets think about the tangents to our surface at point A, in the x- and y-directions. In this view, looking down on the surface, these tangents will be colinear with the -i and -j vectors in the diagram but they will be angled upwards. Let's call these vectors p (in the x-direction) and q (in the y-direction). The vector sum p+q will give us the tangent vector in the x=y direction, which we'll call vector u. 

②   What is the magnitude of the z-component of vector u ?

③  What is the gradient of vector u ?