Finding shortest distance from a point to line through direction vector

Question

Find the shortest distance from a point $P(1,3,-2)$ to the line through $P_0 (2,0,-1)$ with direction vector $d = (1, -1, 0)$.

I know how to find distance between a point $(x,y)$ and a line $ax+by+c=0$ but I have no idea how to find it through another point and what & how to use direction vector.

Answer 1

If you want to use cross-product: as you know, $$|a\times b|=|a||b|\sin\theta$$ where $\theta$ is the angle between vectors $a$ and $b$

The shortest distance you require is $$|PP_0|\sin\theta$$ where $\theta$ is the angle between the vector $PP_0$ and the direction vector of the line.

Can you take it from there?

Answer 2

The squared distance to any point on the line is

$$(2+t-1)^2+(-t-3)^2+(-1+2)^2=2(t+2)^2+3$$ so the shortest distance is $\sqrt3$ (when $t=-2$).