Question

Consider the function f(x,y)=(e^x-4x)sin(y). Suppose S is the surface z=f(x,y). Suppose v=3i+5j+ak is a vector in 3-space which is tangent to the surface S at the point P lying on the surface above (3,3). What is a?

Answer 1

find the normal vectors taking gradient of G(x,y,z) = f(x,y) - z
http://upload.wikimedia.org/wikipedia/en/math/8/8/4/8842f1f40ec25ebb86edd2a4659a9ebb.png
the dot product of this normal vector with v=3i+5j+ak must be zero.

Answer 2

If you solve what @experimnetX suggested,
you get
\[
a=-12 \sin (3)+3 e^3 \sin (3)-60 \cos (3)+5 e^3 \cos (3)
\]

Answer 3

NUmerically
a=-33.2131