What is the unit vector with positive z component which is normal to z = xy + x/y at (1,1,2)?

1 Answer
Dec 3, 2015

Answer:

This is really a calculus question, but the answer is #vec(n)=<-2/sqrt(5),0,1/sqrt(5)> = -2/sqrt(5) hat(i)+1/sqrt(5) hat(k)#.

Explanation:

Rewrite the equation #z=xy+x/y# as #xy+x/y-z=0# and let #F(x,y,z)=xy+x/y-z# so that the equation is #F(x,y,z)=0#.

The gradient vector of #F# is normal to the surface at any given point on the surface. That gradient vector is

#nabla F=< (partial F)/(partial x), (partial F)/(partial y), (partial F)/(partial z)> = < y+1/y,x-x/y^2,-1>#.

At the point #(x,y,z)=(1,1,2)# (which you should check is on the original surface), this gradient vector is #nabla F=<2,0,-1>#.

Since #||nabla F||=sqrt{2^{2}+0^{2}+(-1)^{2}}=sqrt{5}#, it follows that #vec(n)=-(nabla F)/||nabla F||=<-2/sqrt(5),0,1/sqrt(5)> = -2/sqrt(5) hat(i)+1/sqrt(5) hat(k)# is a unit vector normal to this surface with a positive #z#-component.

Here's a picture of this situation:

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