How do I find a tangent plane without a specified point?

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I was having a problem finding the points on $z=3x^2 - 4y^2$ where vector $n=<3,2,2>$ is normal to the tangent plane.



How do we calculate the tangent plane equation without a specific point to calculate it at?



I also had an idea to take the cross product of $2$ vectors in the plane and somehow compare it to the $n$ vector but I don't know exactly how to do this. Thank you for any help!






multivariable-calculus vectors orthonormal tangent-spaces







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  • Set $x=0$ and $y=0$ in the eqution anc compute $z$. This is a point.
    – Mauro ALLEGRANZA
    2 hours ago











  • I don't think that is exactly what the OP is asking!
    – user247327
    1 hour ago














up vote
2
down vote

favorite
2












I was having a problem finding the points on $z=3x^2 - 4y^2$ where vector $n=<3,2,2>$ is normal to the tangent plane.



How do we calculate the tangent plane equation without a specific point to calculate it at?



I also had an idea to take the cross product of $2$ vectors in the plane and somehow compare it to the $n$ vector but I don't know exactly how to do this. Thank you for any help!






multivariable-calculus vectors orthonormal tangent-spaces







share|cite|improve this question























  • Set $x=0$ and $y=0$ in the eqution anc compute $z$. This is a point.
    – Mauro ALLEGRANZA
    2 hours ago











  • I don't think that is exactly what the OP is asking!
    – user247327
    1 hour ago












up vote
2
down vote

favorite
2









up vote
2
down vote

favorite
2






2





I was having a problem finding the points on $z=3x^2 - 4y^2$ where vector $n=<3,2,2>$ is normal to the tangent plane.



How do we calculate the tangent plane equation without a specific point to calculate it at?



I also had an idea to take the cross product of $2$ vectors in the plane and somehow compare it to the $n$ vector but I don't know exactly how to do this. Thank you for any help!






multivariable-calculus vectors orthonormal tangent-spaces







share|cite|improve this question















I was having a problem finding the points on $z=3x^2 - 4y^2$ where vector $n=<3,2,2>$ is normal to the tangent plane.



How do we calculate the tangent plane equation without a specific point to calculate it at?



I also had an idea to take the cross product of $2$ vectors in the plane and somehow compare it to the $n$ vector but I don't know exactly how to do this. Thank you for any help!






multivariable-calculus vectors orthonormal tangent-spaces




multivariable-calculus vectors orthonormal tangent-spaces






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share|cite|improve this question













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edited 1 hour ago









hamza boulahia

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asked 2 hours ago









sjfklsdafjks

1132




1132











  • Set $x=0$ and $y=0$ in the eqution anc compute $z$. This is a point.
    – Mauro ALLEGRANZA
    2 hours ago











  • I don't think that is exactly what the OP is asking!
    – user247327
    1 hour ago
















  • Set $x=0$ and $y=0$ in the eqution anc compute $z$. This is a point.
    – Mauro ALLEGRANZA
    2 hours ago











  • I don't think that is exactly what the OP is asking!
    – user247327
    1 hour ago















Set $x=0$ and $y=0$ in the eqution anc compute $z$. This is a point.
– Mauro ALLEGRANZA
2 hours ago





Set $x=0$ and $y=0$ in the eqution anc compute $z$. This is a point.
– Mauro ALLEGRANZA
2 hours ago













I don't think that is exactly what the OP is asking!
– user247327
1 hour ago




I don't think that is exactly what the OP is asking!
– user247327
1 hour ago










3 Answers
3






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up vote
3
down vote



accepted










Let $f(x,y,z)=3x^2-4y^2-z$. Then your surface is $bigl(x,y,z)inmathbbR^3,$. You are after the points $(x,y,z)$ in that surface such that $nabla f(x,y,z)$ is a multiple of $(3,2,2)$. So, solve the system$$left{beginarrayl6x=3lambda\-8y=2lambda\-1=2lambda\3x^2-4y^2-z=0.endarrayright.$$






share|cite|improve this answer




















  • Just wanted to clarify for others that I used the bottom-most equation after getting x and y from the top 2 equations and lambda from the third one. Thank you for the answer!!
    – sjfklsdafjks
    1 hour ago










  • I'm glad I could help.
    – José Carlos Santos
    1 hour ago

















up vote
3
down vote













Let $ f(x,y,z)=3x^2 - 4y^2 - z$ then the normal vector at $p_0(x_0,y_0,z_0)$ is
$$nabla f(p)=(f_x,f_y,f_z)_p$$
or
$$nabla f(p)=(6x_0,-8y_0,-1)$$
then
$$dfracnabla f(p)=dfracvecnvecn$$






share|cite|improve this answer




















  • Be aware that the signs of the vectors could differ!
    – weee
    1 hour ago










  • yes, equality with a $pm$.
    – Nosrati
    1 hour ago










  • And then how can we calculate the tangent plane?
    – manooooh
    1 hour ago










  • $p_0=(-frac14,-frac18,frac18)$.
    – Nosrati
    1 hour ago

















up vote
0
down vote













The problem does not ask you to find a tangent plane! It asks you to find points where the normal vector is parallel to <3, 2, 2>. The normal vector at any point of f(x,y,z)= constant is $nabla f$. Here $f(x, y, z)= 3x^2- 4y^2- z= 0$. Find $nabla f$ and set it equal to <3k, 2k, 2k> for some k.






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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    3
    down vote



    accepted










    Let $f(x,y,z)=3x^2-4y^2-z$. Then your surface is $bigl(x,y,z)inmathbbR^3,$. You are after the points $(x,y,z)$ in that surface such that $nabla f(x,y,z)$ is a multiple of $(3,2,2)$. So, solve the system$$left{beginarrayl6x=3lambda\-8y=2lambda\-1=2lambda\3x^2-4y^2-z=0.endarrayright.$$






    share|cite|improve this answer




















    • Just wanted to clarify for others that I used the bottom-most equation after getting x and y from the top 2 equations and lambda from the third one. Thank you for the answer!!
      – sjfklsdafjks
      1 hour ago










    • I'm glad I could help.
      – José Carlos Santos
      1 hour ago














    up vote
    3
    down vote



    accepted










    Let $f(x,y,z)=3x^2-4y^2-z$. Then your surface is $bigl(x,y,z)inmathbbR^3,$. You are after the points $(x,y,z)$ in that surface such that $nabla f(x,y,z)$ is a multiple of $(3,2,2)$. So, solve the system$$left{beginarrayl6x=3lambda\-8y=2lambda\-1=2lambda\3x^2-4y^2-z=0.endarrayright.$$






    share|cite|improve this answer




















    • Just wanted to clarify for others that I used the bottom-most equation after getting x and y from the top 2 equations and lambda from the third one. Thank you for the answer!!
      – sjfklsdafjks
      1 hour ago










    • I'm glad I could help.
      – José Carlos Santos
      1 hour ago












    up vote
    3
    down vote



    accepted







    up vote
    3
    down vote



    accepted






    Let $f(x,y,z)=3x^2-4y^2-z$. Then your surface is $bigl(x,y,z)inmathbbR^3,$. You are after the points $(x,y,z)$ in that surface such that $nabla f(x,y,z)$ is a multiple of $(3,2,2)$. So, solve the system$$left{beginarrayl6x=3lambda\-8y=2lambda\-1=2lambda\3x^2-4y^2-z=0.endarrayright.$$






    share|cite|improve this answer












    Let $f(x,y,z)=3x^2-4y^2-z$. Then your surface is $bigl(x,y,z)inmathbbR^3,$. You are after the points $(x,y,z)$ in that surface such that $nabla f(x,y,z)$ is a multiple of $(3,2,2)$. So, solve the system$$left{beginarrayl6x=3lambda\-8y=2lambda\-1=2lambda\3x^2-4y^2-z=0.endarrayright.$$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 1 hour ago









    José Carlos Santos

    136k17109198




    136k17109198











    • Just wanted to clarify for others that I used the bottom-most equation after getting x and y from the top 2 equations and lambda from the third one. Thank you for the answer!!
      – sjfklsdafjks
      1 hour ago










    • I'm glad I could help.
      – José Carlos Santos
      1 hour ago
















    • Just wanted to clarify for others that I used the bottom-most equation after getting x and y from the top 2 equations and lambda from the third one. Thank you for the answer!!
      – sjfklsdafjks
      1 hour ago










    • I'm glad I could help.
      – José Carlos Santos
      1 hour ago















    Just wanted to clarify for others that I used the bottom-most equation after getting x and y from the top 2 equations and lambda from the third one. Thank you for the answer!!
    – sjfklsdafjks
    1 hour ago




    Just wanted to clarify for others that I used the bottom-most equation after getting x and y from the top 2 equations and lambda from the third one. Thank you for the answer!!
    – sjfklsdafjks
    1 hour ago












    I'm glad I could help.
    – José Carlos Santos
    1 hour ago




    I'm glad I could help.
    – José Carlos Santos
    1 hour ago










    up vote
    3
    down vote













    Let $ f(x,y,z)=3x^2 - 4y^2 - z$ then the normal vector at $p_0(x_0,y_0,z_0)$ is
    $$nabla f(p)=(f_x,f_y,f_z)_p$$
    or
    $$nabla f(p)=(6x_0,-8y_0,-1)$$
    then
    $$dfracnabla f(p)=dfracvecnvecn$$






    share|cite|improve this answer




















    • Be aware that the signs of the vectors could differ!
      – weee
      1 hour ago










    • yes, equality with a $pm$.
      – Nosrati
      1 hour ago










    • And then how can we calculate the tangent plane?
      – manooooh
      1 hour ago










    • $p_0=(-frac14,-frac18,frac18)$.
      – Nosrati
      1 hour ago














    up vote
    3
    down vote













    Let $ f(x,y,z)=3x^2 - 4y^2 - z$ then the normal vector at $p_0(x_0,y_0,z_0)$ is
    $$nabla f(p)=(f_x,f_y,f_z)_p$$
    or
    $$nabla f(p)=(6x_0,-8y_0,-1)$$
    then
    $$dfracnabla f(p)=dfracvecnvecn$$






    share|cite|improve this answer




















    • Be aware that the signs of the vectors could differ!
      – weee
      1 hour ago










    • yes, equality with a $pm$.
      – Nosrati
      1 hour ago










    • And then how can we calculate the tangent plane?
      – manooooh
      1 hour ago










    • $p_0=(-frac14,-frac18,frac18)$.
      – Nosrati
      1 hour ago












    up vote
    3
    down vote










    up vote
    3
    down vote









    Let $ f(x,y,z)=3x^2 - 4y^2 - z$ then the normal vector at $p_0(x_0,y_0,z_0)$ is
    $$nabla f(p)=(f_x,f_y,f_z)_p$$
    or
    $$nabla f(p)=(6x_0,-8y_0,-1)$$
    then
    $$dfracnabla f(p)=dfracvecnvecn$$






    share|cite|improve this answer












    Let $ f(x,y,z)=3x^2 - 4y^2 - z$ then the normal vector at $p_0(x_0,y_0,z_0)$ is
    $$nabla f(p)=(f_x,f_y,f_z)_p$$
    or
    $$nabla f(p)=(6x_0,-8y_0,-1)$$
    then
    $$dfracnabla f(p)=dfracvecnvecn$$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 1 hour ago









    Nosrati

    25.1k62052




    25.1k62052











    • Be aware that the signs of the vectors could differ!
      – weee
      1 hour ago










    • yes, equality with a $pm$.
      – Nosrati
      1 hour ago










    • And then how can we calculate the tangent plane?
      – manooooh
      1 hour ago










    • $p_0=(-frac14,-frac18,frac18)$.
      – Nosrati
      1 hour ago
















    • Be aware that the signs of the vectors could differ!
      – weee
      1 hour ago










    • yes, equality with a $pm$.
      – Nosrati
      1 hour ago










    • And then how can we calculate the tangent plane?
      – manooooh
      1 hour ago










    • $p_0=(-frac14,-frac18,frac18)$.
      – Nosrati
      1 hour ago















    Be aware that the signs of the vectors could differ!
    – weee
    1 hour ago




    Be aware that the signs of the vectors could differ!
    – weee
    1 hour ago












    yes, equality with a $pm$.
    – Nosrati
    1 hour ago




    yes, equality with a $pm$.
    – Nosrati
    1 hour ago












    And then how can we calculate the tangent plane?
    – manooooh
    1 hour ago




    And then how can we calculate the tangent plane?
    – manooooh
    1 hour ago












    $p_0=(-frac14,-frac18,frac18)$.
    – Nosrati
    1 hour ago




    $p_0=(-frac14,-frac18,frac18)$.
    – Nosrati
    1 hour ago










    up vote
    0
    down vote













    The problem does not ask you to find a tangent plane! It asks you to find points where the normal vector is parallel to <3, 2, 2>. The normal vector at any point of f(x,y,z)= constant is $nabla f$. Here $f(x, y, z)= 3x^2- 4y^2- z= 0$. Find $nabla f$ and set it equal to <3k, 2k, 2k> for some k.






    share|cite|improve this answer
























      up vote
      0
      down vote













      The problem does not ask you to find a tangent plane! It asks you to find points where the normal vector is parallel to <3, 2, 2>. The normal vector at any point of f(x,y,z)= constant is $nabla f$. Here $f(x, y, z)= 3x^2- 4y^2- z= 0$. Find $nabla f$ and set it equal to <3k, 2k, 2k> for some k.






      share|cite|improve this answer






















        up vote
        0
        down vote










        up vote
        0
        down vote









        The problem does not ask you to find a tangent plane! It asks you to find points where the normal vector is parallel to <3, 2, 2>. The normal vector at any point of f(x,y,z)= constant is $nabla f$. Here $f(x, y, z)= 3x^2- 4y^2- z= 0$. Find $nabla f$ and set it equal to <3k, 2k, 2k> for some k.






        share|cite|improve this answer












        The problem does not ask you to find a tangent plane! It asks you to find points where the normal vector is parallel to <3, 2, 2>. The normal vector at any point of f(x,y,z)= constant is $nabla f$. Here $f(x, y, z)= 3x^2- 4y^2- z= 0$. Find $nabla f$ and set it equal to <3k, 2k, 2k> for some k.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 1 hour ago









        user247327

        10.2k1515




        10.2k1515



























             

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