Find a point on the line closest to another point

[MarcL]

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?

 

[haruspex]

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?

 

[ehild]

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?

 

[MarcL]

The normal, but... it seems kinda redundant if it lies on the same line , anyway to me at least.

 

[ehild]

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?

 

[Mark44]

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.