Find a unit vector n that is normal to the surface 4z² - x4 - 2y4 = 46 at P = (2, 1, 4) that points in the direction of the xy-plane (in other words, if you travel in the direction of n, you will eventually cross the xy-plane). (Use symbolic notation and fractions where needed.) n=
Find a unit vector n that is normal to the surface 4z² - x4 - 2y4 = 46 at P = (2, 1, 4) that points in the direction of the xy-plane (in other words, if you travel in the direction of n, you will eventually cross the xy-plane). (Use symbolic notation and fractions where needed.) n=
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
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![### Problem Statement:
Find a unit vector **n** that is normal to the surface \( 4z^2 - x^4 - 2y^4 = 46 \) at \( P = (2, 1, 4) \) that points in the direction of the \( xy \)-plane (in other words, if you travel in the direction of **n**, you will eventually cross the \( xy \)-plane).
(Use symbolic notation and fractions where needed.)
### Solution:
\[ \mathbf{n} = \boxed{\phantom{\frac{1}{1}}} \]
---
This problem involves finding a normal vector to a given surface at a specific point and ensuring that the vector points in a specified direction. The main objective is to determine the unit vector **n** that is perpendicular to the surface described by the implicit function \( F(x, y, z) = 4z^2 - x^4 - 2y^4 - 46 \).
To achieve this, you should apply the concept of gradients and vector normalization in multivariable calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faccc7427-2c55-44a5-b68e-19581b7568be%2Fdf463091-e44d-40a8-9770-fb1d63dc7cbe%2Fx7t30rc_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Find a unit vector **n** that is normal to the surface \( 4z^2 - x^4 - 2y^4 = 46 \) at \( P = (2, 1, 4) \) that points in the direction of the \( xy \)-plane (in other words, if you travel in the direction of **n**, you will eventually cross the \( xy \)-plane).
(Use symbolic notation and fractions where needed.)
### Solution:
\[ \mathbf{n} = \boxed{\phantom{\frac{1}{1}}} \]
---
This problem involves finding a normal vector to a given surface at a specific point and ensuring that the vector points in a specified direction. The main objective is to determine the unit vector **n** that is perpendicular to the surface described by the implicit function \( F(x, y, z) = 4z^2 - x^4 - 2y^4 - 46 \).
To achieve this, you should apply the concept of gradients and vector normalization in multivariable calculus.
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