Explanation Given: The plane ln ( x y − z ) = 0 Concept Used: Gradient of a function f ( x , y , z ) at a point P ( x 0 , y 0 , z 0 ) is: ∇ f = f x ( x 0 , y 0 , z 0 ) i + f y ( x 0 , y 0 , z 0 ) j + f z ( x 0 , y 0 , z 0 ) k The standard equation of tangent plane to a surface S with equation f ( x , y , z ) = C at point P ( x 0 , y 0 , z 0 ) is f x ( x 0 , y 0 , z 0 ) ( x − x 0 ) + f y ( x 0 , y 0 , z 0 ) ( y − y 0 ) + f z ( x 0 , y 0 , z 0 ) ( z − z 0 ) = 0 Calculation: To find a unit vector, normal to the surface ln ( x y − z ) = 0 at ( 2 , 5 , 3 ) First, convert the equation of surface in more simplified form as ln ( x y − z ) = 0 ⇒ x y − z = e 0 ⇒ x y − z = 1 ⇒ x = y − z ⇒ x − y + z = 0 Let, f ( x , y , z ) = x − y + z Now, find the gradient vector of this function at the point ( 2 , 5 , 3 ) , since gradient vector at any point is perpendicular to the surface at that point. Gradient of a function f ( x , y , z ) is ∇ f = f x ( x 0 , y 0 , z 0 ) i + f y ( x 0 , y 0 , z 0 ) j + f z ( x 0 , y 0 , z 0 ) k Thus, first find f x ( 2 , 5 , 3 ) , f y ( 2 , 5 , 3 ) , and f z ( 2 , 5 , 3 ) as f x ( x , y , z ) = ∂ ∂ x ( x − y + z ) = ∂ ∂ x x − ∂ ∂ x y + ∂ ∂ x z = 1 − 0 + 0 = 1 , f y ( x , y , z ) = ∂ ∂ y ( x − y + z ) = ∂ ∂ y x −

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 11.6, Problem 21PS

To determine

To find a unit vector that is normal to the surface

ln(xy−z)=0 at (2,5,3)

And to find the standard form of the equation of tangent plane to this surface at above point

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Chapter 11.6, Problem 20PS

Chapter 11.6, Problem 22PS

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To find a unit vector that is normal to the surface ln ( x y − z ) = 0 at ( 2 , 5 , 3 ) And to find the standard form of the equation of tangent plane to this surface at above point

Solution Summary: The author explains how to find a unit vector that is normal to the surface and the standard form of the equation of tangent plane to this surface.

Videos

To find a unit vector that is normal to the surface

ln(xy−z)=0 at (2,5,3)

And to find the standard form of the equation of tangent plane to this surface at above point

Want to see the full answer?

Chapter 11 Solutions

To find a unit vector that is normal to the surface ln ( x y − z ) = 0 at ( 2 , 5 , 3 ) And to find the standard form of the equation of tangent plane to this surface at above point

Solution Summary: The author explains how to find a unit vector that is normal to the surface and the standard form of the equation of tangent plane to this surface.

Videos

To find a unit vector that is normal to the surface ln(xy−z)=0 at (2,5,3) And to find the standard form of the equation of tangent plane to this surface at above point

Want to see the full answer?

Chapter 11 Solutions

To find a unit vector that is normal to the surface

ln(xy−z)=0 at (2,5,3)

And to find the standard form of the equation of tangent plane to this surface at above point