Slader Find a unit vector n that is normal to the surface z 2 − 2 x 4 − y 4 = 16 at P = ( 2 , 2 , 8 ) that points in the direction of the x y -plane (in other words, if you travel in the direction of n , you will eventually cross the x y -plane).

u = 0.873 i + 0.436 j - 0.218 k

Step-by-step explanation:

Given:

- The given surface as follows:

                          z^2 - 2*x^4 - y^4 = 16

- The point P = ( 2 , 2 , 8 )

Find:

Find a unit vector n that is normal to the surface at point P such that it points in the direction of x-y plane.

Solution:

- Any vector that is normal to a surface in any coordinate system is given by its directional derivative ∀:

- Where ∀ is the partial derivatives of the function with respect to x, y , and z directions as follows:

                                 ∀ = d/dx i + d/dy j + d/dz k

Where, f is the given function as follows:

                                 f ( x , y , z ) = z^2 - 2*x^4 - y^4 - 16

- The directional derivative is given as:

                                 D_f = ∀.f = -8*x^3 i - 4*y^3 j + 2*z

- The above result is the normal vector that at any general point on the surface. So at point P the normal vector n would be:

                                n = -8*2^3 i - 4*2^3 j + 2*8 k

                                n = -64 i - 32 j + 16 k

- Now we have to check the direction of the normal vector whether it passes the x-y plane. For any vector to pass the x-y plane or z=0. Its kth component must be "negative" because as we along the vector the position of each point must be closer to x-y plane or approach z = 0. Hence, the normal vector in the direction of x-y plane is:

                               n = -1*( -64 i - 32 j + 16 k )

                               n = 64 i + 32 j -16 k

- The corresponding unit vector (u) of the normal vector (n) is given as :

                               u = n / | n |

Where, | n | is the magnitude of the normal vector:

                               | n | = sqrt ( 64^2 + 32^2 + 16^2)

                               | n | = sqrt (5376)

                               | n | = 73.3212

- The unit vector is given as:

                                u = 64 i + 32 j -16 k  / 73.3212

                                u = 0.873 i + 0.436 j - 0.218 k