Find an equation for the tangent plane to a surface (using gradient)

[FaraDazed]

Homework Statement

Find an equation for the tangent plane to a surface $xz^2 +x^2y-z=-1$ at the point (1,-3,2).

Homework Equations

$(\vec{r}-\vec{r_p}) \cdot \nabla f(\vec{r_p}) = 0$

The Attempt at a Solution

[/B]
First I found the gradient of the function
$\nabla f = (z^2+2xy)\hat{i} + x^2 \hat{j} + (2xz-1)\hat{k}$
and then evaluated at (1,-3,2)
$\nabla f(\vec{r_p}) = (2^2+2(1)(-3))\hat{i} + 1^2 \hat{j} + (2(1)(2)-1)\hat{k} = (-2,1,3)$

And then used the relation in the relevant equations section:
$(\vec{r}-\vec{r_p}) \cdot \nabla f(\vec{r_p}) = 0 \\ (x-1,y+3,z-2) \cdot (-2,1,3) = 0 \\ (-2x+2)+(y+3)+(3z-6) = 0 \\ -2x+y+3z=1$

This is the first time I've ever done this and I am just going off of a similar example that the solution was given for. It is not exactly the same question so not sure my process is even correct. Would appreciate any help :)

 

[Mark44]

Homework Statement

Find an equation for the tangent plane to a surface $xz^2 +x^2y-z=-1$ at the point (1,-3,2).

Homework Equations

$(\vec{r}-\vec{r_p}) \cdot \nabla f(\vec{r_p}) = 0$

The Attempt at a Solution

[/B]
First I found the gradient of the function
$\nabla f = (z^2+2xy)\hat{i} + x^2 \hat{j} + (2xz-1)\hat{k}$
and then evaluated at (1,-3,2)
$\nabla f(\vec{r_p}) = (2^2+2(1)(-3))\hat{i} + 1^2 \hat{j} + (2(1)(2)-1)\hat{k} = (-2,1,3)$

And then used the relation in the relevant equations section:
$(\vec{r}-\vec{r_p}) \cdot \nabla f(\vec{r_p}) = 0 \\ (x-1,y+3,z-2) \cdot (-2,1,3) = 0 \\ (-2x+2)+(y+3)+(3z-6) = 0 \\ -2x+y+3z=1$

This is the first time I've ever done this and I am just going off of a similar example that the solution was given for. It is not exactly the same question so not sure my process is even correct. Would appreciate any help :)

Looks good to me.

As a check you can do the following:

• Confirm that the normal to the plane is parallel to the gradient at the given point.
• Confirm that the plane contains the given point.

 

[FaraDazed]

Looks good to me.

As a check you can do the following:

• Confirm that the normal to the plane is parallel to the gradient at the given point.
• Confirm that the plane contains the given point.

Thanks for taking a look, and for the advice on how to check if correct, appreciate it :)

 

[HallsofIvy]

Mark44 told you that!
"As a check you can do the following:

• Confirm that the normal to the plane is parallel to the gradient at the given point.
• Confirm that the plane contains the given point."