Chapter 5.2, Problem 30P

Problems

For Problems 29-32, use result of problem $28$ to find the distance from the given point $P$ to the given plane $\mathcal{P}$.

$P\left(0,-1,3\right)$; Plane $P$ with equation $3x-y-z=5$

In this problem, we use the ideas of this section to derive a formula for the distance from a point $P\left(x_0,y_0,z_0\right)$ in ${ℝ}^3$ to a plane $\mathcal{P}$ with equation $ax+by+cz+d=0$. Unlike the situation of the distance from a point to a line in which we performed orthogonal projection onto a vector pointing along the line followed by vector subtraction, here we can project directly onto a normal vector $n=\left(a,b,c\right)$ for the plane without the need for vector subtraction.

(a) Draw a picture of this situation, including the point $P$, the plane $\mathcal{P}$, and the normal vector $n$ to the plane.

(b) Choosing a point $Q\left(x_1.y_1,z_1\right)$ on the plane, construct the vector $w$ from $Q$ to $P$ and show geo-metrically that the distance from $P\left(x_0,y_0,z_0\right)$ to the plane is $\Vert P\left(w,n\right)\Vert$.

(c) Find a formula for $\Vert P\left(w,n\right)\Vert$ in terms of $a$, $b$, $c$, $d$, $x_0$, $y_0$ and $z_0$.