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)?

Question

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)?

Jayden Landry · Accepted Answer

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                              }

How to find a unit vector normal to the surface x 3 + y 3...

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