Question

Find the coordinates of the point (x, y, z) on the plane z = 3 x + 4 y + 4 which is closest to the origin.

Answer 1

okay so 3-d is very similar to 2-d

Answer 2

what point on that line will be closest to the origin?

Answer 3

you can label it using the edit drawing button if you would like

Answer 4

What is the formula for distance from the origin to just some arbitrary point, (x,y,z)?
We would use that formula with the constraint that "z = 3 x + 4 y + 4"...
from there, I'm not exactly sure how to the answer from there. I guess you'd use lagrange multipliers but I haven't learned about them yet. :P

Answer 5

my guess would find normal vector, the closest point will be the one that forms a perpendicular line with the origin.

normal = <-3,-4,1>
this represents directional vector through origin
line represented by:
x = -3t
y = -4t
z = t

find point where line intersects plane
z = 3x+4y +4
t = -9t -16t +4
26t = 4
t = 2/13

point --> (-6/13, -8/13, 2/13)

Answer 6

Dumbcow, how did you get t=-9t -16t +4?

Answer 7

im guessing around there?

Answer 8

i substituted in the parametric form of the line into the plane equation to find the point on the line that is also on the plane

Answer 9

okay, I see. Thank you!