Shortest distance from point to plane

• Oct 13th 2011, 04:49 PM
esander4
Shortest distance from point to plane
I don't even know where to start on this one.

"Find the shortest distance from the point (1,0,-2) to the plane x+2y+z=4"
• Oct 13th 2011, 05:04 PM
SammyS
Re: Shortest distance from point to plane
It's the distance along the normal to the plane, x+2y+z=4.

Do you know how to find a vector normal to that plane?
• Oct 13th 2011, 06:43 PM
esander4
Re: Shortest distance from point to plane
Is it the cross product of the partial derivative of x and partial derivative of y? And then the magnitude of that?
• Oct 13th 2011, 11:15 PM
chisigma
Re: Shortest distance from point to plane
Quote:

Originally Posted by esander4
I don't even know where to start on this one.

"Find the shortest distance from the point (1,0,-2) to the plane x+2y+z=4"

The equation of the plane can be written as...

$\displaystyle z(x,y)= 4-x-2 y$ (1)

... so that the squared distance from the point (1,0,-2) is...

$\displaystyle \delta^{2} (x,y)= (x-1)^{2}+ y^{2} + \{z(x,y)+2\}^{2}$ (2)

Now You minimize $\displaystyle \delta^{2}(x,y)$ respect to x and y and the problem is solved...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$