Need help with calc 3... Let P(x0,y0,z0) lie on the unit sphere: x^2+y^2+z^2 = 1. Show that the equation of the plane, (p), containing point P with normal vector OP is: x0x +y0y + z0z = 1 I got as far as assuming point O is (x,y,z) and vector OP to be .

Question

Need help with calc 3...
Let P(x0,y0,z0) lie on the unit sphere: x^2+y^2+z^2 = 1. Show that the equation of the plane, (p), containing point P with normal vector OP is: x0x +y0y + z0z = 1

I got as far as assuming point O is (x,y,z) and vector OP to be <x-x0, y-y0, z-z0>.

anonymous · Answer

@nerd94 @babygirl180

ilovecake · Answer

Do you know what to do?

anonymous · Answer

I think i'm supposed to confirm the equation of the plane with the given point P?

anonymous · Answer

Let $F(x,y,z)=x^2+y^2+z^2$, then
$$\nabla F(x,y,z)\bigg|_{x=x_0,\,y=y_0,\,z=z_0}=\langle2x_0,2y_0,2z_0\rangle$$
is the gradient vector to the surface at $(x_0,y_0,z_0)$.

The tangent plane is described by the equation,
$$\begin{align*}\langle2x_0,2y_0,2z_0\rangle\bullet\langle x-x_0,y-y_0,z-z_0\rangle&=0\\
x_0(x-x_0)+y_0(y-y_0)+z_0(z-z_0)&=0
\end{align*}$$