"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?
Best Answer
The shortest distance should intuitively be the length of the segment which connects our point to our line and forms a right angle. Consider that we can determine this first by determining a vector from a point on our line to $A$ followed by finding its projection onto $\vec{PQ}$; the orthogonal part is then just the difference between this vector and our original vector from our line to our point and its magnitude yields the length of said segment:$$d=\left\|\vec{AP}-\frac{\vec{AP}\cdot\vec{PQ}}{\|\vec{PQ}\|^2}\vec{PQ}\right\|$$