Finding the closest point on a plane to a point not on the plane

[Subdot]

Homework Statement

Find the point P on the plane given by 5x - 14y + 2z + 9 = 0 which is nearest to the point Q = (-2, 15, -7).

Homework Equations

Distance formulas for distance between the plane and the origin and the plane and a point not on a plane. And of course, the Cartesian equation of a plane.

The Attempt at a Solution

I thought the closest point to Q must be along the normal vector to the plane. The vector would be when P - Q is a scalar multiple of N. My problem is that the answer given was (-1/25)(5, -14, 2). I don't understand this answer since it's independent of Q. Is the answer wrong?

 

[arildno]

Remember that any point on the normal joining plane P and point Q can be written as:

p(t)=(5,-14,2)t+(-2,15,-7)

Since we are interested in the value t* making p(t*) a point on the plane, we insert this expression into the equation for P, and solve for t*.

Then, we determine p(t*) explicitly.

 

[Subdot]

I think that's similar to what I tried doing but didn't have time to write out when I was posting that. I'm not sure what you mean by the "equation for P", but I'm guessing you mean the equation for the plane? So what you're saying is, p(t) = tN + Q, P = p(t*) = (t*)N + Q. If so, that is what I tried doing.

I'll carry it through to completion now though. P = (5, -14, 2)t* + (-2, 15, -7), so x = 5t* - 2, y = -14t* + 15, and z = 2t* - 7. Plugging this into the equation for the plane gives 5(5t* - 2) -14(-14t* + 15) + 2(2t* - 7) = 25t* + 196t* + 4t* - 10 - 210 - 14 = 225t* - 234 = -9 --> 225t* = 225 --> t* = 1. Thus, P = N + Q = (3, 1, -5).

Such was the answer I had gotten earlier, yet it does not match up with the book's answer. So did I misunderstand something/do something wrong?

 

[arildno]

The point given as an answer does not, as far as I can see, lie on the plane.

Yours is correct.

 

[Subdot]

Thanks a lot! For future reference, I made a typo in the OP which I have now corrected. The point in the OP is indeed on the plane, but if you are getting the same answer as me on this, then I'm sure that point is incorrect too. If not, anyone please correct this since I only reluctantly change an answer key.