Consider the function F(x,y,z)=x^2*y^3*z^3 Find the normal vector to the level s
ID: 2986959 • Letter: C
Question
Consider the function F(x,y,z)=x^2*y^3*z^3
Find the normal vector to the level surface F(x,y,z)=4 at the point (2,1,1)class="chg-suggest-term">
class="txt-hdr-sec">class="chg-suggest-term">class="txt-hdr-sec">Explanation / Answer
Normal vector for any surface F at the point (x,y,z) is of the form
F_x(x,y,z) i + F_y(x,y,z) j + F_z(x,y,z) k
where F_x means the partial derivative of F with respect to x, same meaning applies for F_y and F_z.
Normal vector at any point (x,y,z) for the given function is of the form
2*x*y^3*z^3 i + 3*x^2*y^2*z^3 j + 3*x^2*y^3*z^2 k
Therefore the normal vector for the surface at (2,1,1) will be
4 i + 12 j + 12 k.
or in general the normal vector can be any multiple of 4 i + 12 j + 12 k which inturn is a multiple of 1 i + 3 j + 3 k, therefore the answer is
p i + 3p j + 3p k where p is any real number.