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Find a point on the line closest to another point

  • Thread starter MarcL
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  • #1
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Homework Statement


(2 part problem) a) A plane passes through the point P(3,1,4) and is orthogona to the line (x-1)/2 = (2-y)/-7 = z-3
b) Find the point on the line closest to point (3,1,4)

Homework Equations


Symmetric equation --> (x-x1)/t = (y-y1)/t = (z-z1)/t ( anyway i think that's what it is

The Attempt at a Solution


I found the equation of the plane using the direction vector as (2,-7,1)
and used this form 2(x-3)-7(y-1)+(z-4)= 0

I can't seem to be able to go on to start b. I would think of plugging P in but that seems way too easy.

Any idea on how I can approach this?
 

Answers and Replies

  • #2
haruspex
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I found the equation of the plane using the direction vector as (2,-7,1)
and used this form 2(x-3)-7(y-1)+(z-4)= 0

I can't seem to be able to go on to start b. I would think of plugging P in but that seems way too easy.

Any idea on how I can approach this?
Where will that closest point be in relation to the plane?
 
  • #3
ehild
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The Attempt at a Solution


I found the equation of the plane using the direction vector as (2,-7,1)
Check the sign of the y component of the direction vector.

and used this form 2(x-3)-7(y-1)+(z-4)= 0
I can't seem to be able to go on to start b. I would think of plugging P in but that seems way too easy.

Any idea on how I can approach this?
The distance of P from the normal line is the length of the line drawn perpendicularly to the normal...Where is that line?
 
  • #4
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The normal, but... it seems kinda redundant if it lies on the same line , anyway to me at least.
 
  • #5
ehild
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The normal, but... it seems kinda redundant if it lies on the same line , anyway to me at least.
Of course, it intersects the normal, but what is the position of the line drawn from P and perpendicular to the normal, with respect to the plane? Try to draw a picture.

If you can not see it, write the distance of any point of the normal line from point P. When is it minimum, and what is that minimum distance?
 
  • #6
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Drawing a picture, as ehild suggested, is an excellent idea. Having an image to look at gives you insights that formulas and equations simply can't provide. This took me a couple of minutes to draw.
Plane_and_Line.png
 

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