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

1. Oct 8, 2015

1. The problem statement, all variables and given/known data
Find an equation for the tangent plane to a surface $xz^2 +x^2y-z=-1$ at the point (1,-3,2).

2. Relevant equations
$(\vec{r}-\vec{r_p}) \cdot \nabla f(\vec{r_p}) = 0$

3. The attempt at a solution

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 :)

Last edited: Oct 8, 2015
2. Oct 8, 2015

### Staff: Mentor

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.

3. Oct 8, 2015

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

4. Oct 9, 2015

### HallsofIvy

Staff Emeritus
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."