distances measured in space

Question

how do we find the distance of a point from a given line measured parallel to a given plane? Here is a a sample question : find a distance of point (2, 3, 4) from line (x+3)/3=(y-2)/6=z/2 measured parallel to plance 3x+2y+2z+5=0

Answer 1

Let's denote the given point as $P$ and the given line as $L(t)$ for the ease of discussion. This direction the distance is measured in needs to be parallel to a given plane. So, the whole problem becomes whether we can find a point $Q$ on $L(t)$ so that the line defined by $P$ and $Q$ is parallel to the given plane. Mathematically, we can solve the problem as

Let $Q = R + t_0\vec{n}$ ,

where R is any point that is already on $L(t)$ and $\vec{n}$ is the direction of $L(t)$.

Then, by solving $\vec{PQ} \cdot \vec{N} = 0 $, where $\vec{N}$ is the normal vector of the given plane, we can obtain the value for $t_0$ and therefore the coordinates for $Q$.

With the particular example in the original post, we can find that point $Q$ is located at (0, 8, 2) and the distance from (2, 3, 4) to it is $\sqrt{33}$.