# Find a point in a line closest to another point

1. Nov 9, 2009

### fireb

1. The problem statement, all variables and given/known data
Given p(4,2,3), Q(5,3,1), R(3,-5,5) and let L be a line that passes through P and Q
Find the coordinates of that point of the line L which is closest to the point R.

2. Relevant equations

3. The attempt at a solution
I found the equation of the line L= (x=4+t, y=2+t, z=3-2t) then i am unsure on how to find the coordinates closest to R.
Any hints on solving this will be appreciated.

2. Nov 9, 2009

### lanedance

the vector from the closest point of the line to the point will be perpindicular to the line (why?)

3. Nov 9, 2009

### fireb

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I think i got it now...

So if we let A(x,y,z) be the closest point from L to R. This condition must be satisfy: RA dot PQ= 0. SO RA = (3-x,-5-y,5-z).
And from the dot product, i get x+y-2z=-12. So, i sub in the L into the equation I just got and solve for t, and equal t=-2, yields x=2
y=0,z=7. So A is (2,0,7)?
Can someone correct me if i am wrong

Thanks

Last edited: Nov 9, 2009
4. Nov 9, 2009

### lanedance

if vectors a & b are perpindicular then:
a x b is a non-zero vector
a . b is zero...