Exercise 39

In this problem, we explore how we can use what we know about vectors and projections to find the distance from a point to a plane.

Let $p$ be the plane with equation $z=-4x+3y+4$, and let $Q = (4,-1,8)$.

1. Show that $Q$ does not lie in the plane $p$.

2. Find a normal vector $\vn$ to the plane $p$.

3. Find the coordinates of a point $P$ in $p$.

4. Find the components of $\overrightarrow{PQ}$. Draw a picture to illustrate the objects found so far.

5. Explain why $|\comp_{\vn} \overrightarrow{PQ}|$ gives the distance from the point $Q$ to the plane $p$. Find this distance.