find vector perpendicular level curve

Question

Consider the function f(x,y) = (e^x - 2x)sin(y). Suppose S is the surface z = f (x,y)

A. Find a vector which is perpendicular to the level curve of f through the point (4,3) in the direction in which f decrease most rapidly.

B. Suppose v - 7i + 4j + ak is a vector in 3-space which is tangent to the surface S at the point P lying on the surface above (4, 3). What is a?

Answer 1

For A, we know that the gradient vector is perpendicular to the level curve

(e^x-2x)*sin(y) = f(4,3). So, evaluate the negative gradient (direction of maximum decrease) at the point (4,3).

For B, We can infer that this vector v is perpendicular to the gradient at (4,3). So, set up the equation

v dot gradf(4,3) = 0 and solve for a.