Question

Prove Theorem 10.40. That is, show that if $P=\left(x_0,y_0,z_0\right)$is a point on plane$\wp$then the distance between the origin and $\wp$is given by where $n$is a unit normal vector to $\wp$

Short answer

The distance between the origin and the plane $\rho$is given by where $\mathbf{n}$is a unit normal vector to the plane $\wp$

Step-by-step solution

Given information

The point $P=\left(x_0,y_0,z_0\right)$on the plane $\wp$

Calculation

The goal is to demonstrate that the distance between the origin and the plane is equal $\wp$is given by , where $\mathbf{n}$is a unit normal vector to the plane $\rho .$

The direction vector of the point $P=\left(x_0,y_0,z_0\right)$and the origin $(0,0,0)$is:

Calculation

The distance is given by:

Therefore, the distance between the origin and the plane $\rho$ is given by , where $\mathbf{n}$is a unit normal vector to the plane $\wp$