Question

Find the point on the plane 2x -3y +z = 3 that is closest to the origin

so I picked the point P(1,0,1). then I use a projection? and something else?

Answer 1

I can only do this problem with lagrange multipliers, which is a multi variable calc tehnique

Answer 2

too bad I think Im getting it now.. Im in linear algebra.

Answer 3

It can be done using gradient
Find the square distance from (0,0,0} to {x,y,3-2x +3y
Minimize it (Solve for gradient ={0,0,0}
You are done

Answer 4

You end up needing to minimize the function
\[
f(x,y)=5 x^2-12 x y-12 x+10 y^2+18 y+9
\]

Answer 5

I got (3/7,-9/14,3/14) does this look correct?

Answer 6

@eliassaab won't that require multi variable calc techniques like lagrange multipliers?

Answer 7

\[
gadienr(f)={-12 + 10 x - 12 y, 18 - 12 x + 20 y, 0}
\]
You find where the gradient is zero at
\[
\left\{\left\{x\to \frac{3}{7},y\to -\frac{9}{14}\right\}\right\}
\]

Answer 8

Yes, it is correct

Answer 9

\[
grad(f)=\{10 x-12 y-12,-12 x+20 y+18,0\}
\]

Answer 10

great thank you!. and the distance would be the sqrt((x2-x1)^2 +(y2-y1)^2 + (z2-z1)^2)?

Answer 11

It only requires the use of the gradient.
Lagrange Multipliers does use gradient technique. You can use it in this problem if you want, but you do not have to. Of course, we are using multi-variable calculus/

Answer 12

It is passed my bedtime, I am going to sleep.