How to find a unit vector normal to the surface x^3+y^3+3xyz=3 ay the point(1,2,-1)?

Jaelyn Mueller

Jaelyn Mueller

Answered question

2023-02-18

How to find a unit vector normal to the surface x 3 + y 3 + 3 x y z = 3 ay the point(1,2,-1)?

Answer & Explanation

Jayden Landry

Jayden Landry

Beginner2023-02-19Added 7 answers

f ( x , y , z ) = x 3 + y 3 + 3 x y z - 3 = 0
The gradient of f ( x , y , z ) at point x , y , z is a vector normal to the surface at this point.
Following is how the gradient is obtained
f ( x , y , z ) = ( f x , f y , f z ) = 3 ( x 2 + y z , y 2 + x z , x y ) at point
( 1 , 2 , - 1 ) has the value
3 ( - 1 , 3 , 2 ) and the unit vector is
{ - 1 , 3 , 2 } 1 + 3 2 + 2 2 = { - 1 14 , 3 14 , 2 7 }

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