Find the distance from the point P to a line - linear algebra

[Oliviacarone]

Homework Statement

Find the distance from point P (1,7,3) to the line
(x,y,z) = (-2,1,4) + s(1,-3,4),
s is a free variable

Homework Equations

proj_nQP = ( QP⋅n/(lengthQP)(lengthn) )(n)

The Attempt at a Solution

I'm not quite sure about how to find the normal (n) here, but if I make s=0, I'm guessing it's just (-2,1,4)?
QP would be Q= a point on the line, so I'm not sure which point to use since all I have here is a point, it feels like, and I used this point (-2,1,4) for the normal...
Then if I found QP I would just plug in everything.

 

[tnich]

I'm not quite sure about how to find the normal (n) here, but if I make s=0, I'm guessing it's just (-2,1,4)?

I think the problem is that you are guessing. Try sketching the line and the point. When you understand how parametric representations of lines work, you can start to solve problems with them.

 

[Ray Vickson]

Homework Statement

Find the distance from point P (1,7,3) to the line
(x,y,z) = (-2,1,4) + s(1,-3,4),
s is a free variable

Homework Equations

proj_nQP = ( QP⋅n/(lengthQP)(lengthn) )(n)

The Attempt at a Solution

I'm not quite sure about how to find the normal (n) here, but if I make s=0, I'm guessing it's just (-2,1,4)?
QP would be Q= a point on the line, so I'm not sure which point to use since all I have here is a point, it feels like, and I used this point (-2,1,4) for the normal...
Then if I found QP I would just plug in everything.

You could use calculus to minimize the squared distance between P and a point Q in the line; that squared-distance is a quadratic form in the variable $s$.