Answer:
our first questions about the steepness of a surface: starting at a point on a surface given by f(x,y)f(x,y), and walking in a particular direction, how steep is the surface? We are now ready to answer the question.
We already know roughly what has to be done: as shown in figure 14.3.1, we extend a line in the xx-yy plane to a vertical plane, and we then compute the slope of the curve that is the cross-section of the surface in that plane. The major stumbling block is that what appears in this plane to be the horizontal axis, namely the line in the xx-yy plane, is not an actual axis—we know nothing about the "units'' along the axis. Our goal is to make this line into a tt axis; then we need formulas to write xx and yy in terms of this new variable tt; then we can write zz in terms of tt since we know zz in terms of xx and yy; and finally we can simply take the derivative.
So we need to somehow "mark off'' units on the line, and we need a convenient way to refer to the line in calculations. It turns out that we can accomplish both by using the vector form of a line.♡