Find the shortest distance between the point $(8,3,2)$ and the line through the points $(1,2,1)$ and $(0,4,0)$ "Find the shortest distance between the point $(8,3,2)$ and the line through the points $(1,2,1)$ and $(0,4,0)$"
$$P = (1,2,1), Q = (0,4,0), A = (8,3,2)$$
$OP$ = vector to $P$
$$PQ_ = (0,4,0) - (1,2,1)$$
I found that the equation of the line $L$ that passes through $(1,2,1)$ and $(0,4,0)$ is: 
$$L = OP + PQ \, t;$$
$$L = (1,2,1) + (-1,2,-1) \, t .$$
However after this I'm not sure how to proceed. I can find PA_ then draw a line from $A$ to the line $L$... advice?

Here is an approach, we have $P = (1,2,1), Q = (0,4,0), A = (8,3,2)$
1) Construct a vector $u=AP$
2) Construct a vector $v=AQ$
3) Find the projection $p$ of $u$ on $v$
4) Construct the vector $w = \frac {p}{||v||}v$
5) Construct the vector $d=u-w$ which its norm ||d|| gives the desired result.   
