I have a plane: 1 x + -5 y + 5 z = -7

and a line: (x,y,z) = (0,0,0) + t(-1,-3,2)

and i need to find thesmallest distance between the two.

My work:

Plane : normal = (1,-5,5) & point P = (-7,0,0)

Line : direction vector = (-1,-3,2) & point Q = (0,0,0)

Need to find vector PQ = (7,0,0)

The distance = ||Projection of PQ onto normal n ||

( ( (7,0,0)(1,-5,5) ) / ||(1,-5,5)|| ) * (1,-5,5)

= (7/51) * sqrt51 or 7/sqrt51

This is wrong, and after looking online and through my textbook, i think it’s because I’m not getting the shortest distance, but rather the distance to P

I can’t get a hold of my prof and the help room at my school thought it looked fine but it isn’t right

I’ve searched online for the past three days and i just KNOW this type of question will be on a test

I need to know how to do this, please help