Equation of a Tangent Plane

Question

Find the equation of the tangent plane to the given surface at the given point.

$x=u^2, y=v^2, z=uv$ at $u=1, v=1$

How would you find the tangent plane when the surface is in this format?

Thanks.

score 3 · Accepted Answer · answered Jul 22 '14 at 18:24

3

The tangent vectors to the surface are $$ \mathbf{t}_u=\left(\frac{\partial x}{\partial u},\frac{\partial y}{\partial u},\frac{\partial z}{\partial u}\right)\\ \mathbf{t}_v=\left(\frac{\partial x}{\partial v},\frac{\partial y}{\partial v},\frac{\partial z}{\partial v}\right) $$ The normal to the surface is $\mathbf{n}=\mathbf{t}_u\times\mathbf{t}_v$.
The equation of the plane is $\mathbf{n}\cdot(P-P_0)=0$.

Are this elements sufficient to build your answer?

answered Jul 22 '14 at 18:24

Vincenzo Tibullo

10,955

What are $P$ and $P_0$? Thanks for your answer – user7000 Jul 22 '14 at 18:38
A generic $(x,y,z)$ for points on the plane, and some specific point you know the plane contains. – 2'5 9'2 Jul 22 '14 at 18:43
As said, $P=(x,y,z)$ and $P_0=(x_0,y_0,z_0)$, where $x_0=u_0^2=1$, $y_0=v_0^2=1$ and $z_0=u_0v_0=1$. – Vincenzo Tibullo Jul 22 '14 at 19:07
So the final answer will be in terms of $x,y,$ and $z$? Should the $u$ and $v$ terms of $\textbf{n}$ be converted to $x,y,z$ terms? Thank you @enzotib, alex.jordan. – user7000 Jul 22 '14 at 19:45
@user7000: $\mathbf{n}$ should be evaluated at $P_0$, so $u,v$ should be substituted with $u_0=1$ and $v_0=1$. – Vincenzo Tibullo Jul 22 '14 at 20:43

score 2 · Answer 2 · answered Jul 22 '14 at 18:29

An alternative to @enzotib's answer is to see if you can find an implicit equation for the surface and use the gradient. In this case, $z^2=xy$, so with $F(x,y,z)=xy-z^2$, $\nabla F=\langle F_x,F_y,F_z\rangle$ evaluated at $(1,1,1)$ will give a normal vector to the plane. As with @enzotib's answer, $\vec{n}\cdot(P-P_0)=0$ gives the equation of the tangent plane.

Note that the surface $z^2=xy$ is not exactly the same as the original one, since the original surface has only one sheet, and this implicit equation has a second sheet with $x,y<0$. But at $u=1,v=1$, which is at $(1,1,1)$, this is irrelevant.

This approach only generalizes if you are successfully able to represent the parametrized surface with an implicit equation.

Equation of a Tangent Plane

2 Answers2