Find the points on the surface closest to the origin

In summary, The conversation is about finding the points on the surface z^2=xy+4 closest to the origin. One person suggests using the distance formula, but it becomes messy. Another person suggests using Lagrange multipliers and provides a link for more information. Another person suggests minimizing x^2+ y^2+ z^2 instead of \sqrt{x^2+ y^2+ z^2} to avoid messiness.
  • #1
EgyptMind
3
0
1. Homework Statement
find the points on the surface z^2=xy+4 closest to the origin

i tried the distance formula but it got too messy, i think there's an approach using lagrange multiplier but i just can't quiet figure it out...?
 
Last edited:
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  • #2
to use lagrange, write the function you are trying to minimise (distance to origin r^2 = x^2 + y^2 + z^2), and use the surface as the constraint
 
  • #3
a little more details...?
 
  • #4
what have you tried? post #2 outlines a method to attempt
 
  • #5
if you;re not sure about lagrange multipliers, try a google search, i found
http://www.uAlberta.ca/MATH/gauss/fcm/calculus/multvrbl/basic/Extrema/Lgrng_mltplrs_exmpl.htm
 
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  • #6
EgyptMind said:
1. Homework Statement
find the points on the surface z^2=xy+4 closest to the origin

i tried the distance formula but it got too messy, i think there's an approach using lagrange multiplier but i just can't quiet figure it out...?
What did you try? If you use lanedance's suggestion, to minimize [itex]x^2+ y^2+ z^2[/itex] rather than [itex]\sqrt{x^2+ y^2+ z^2}[/itex] (whatever minimizes one minimizes the other) it should not be "too messy".
 

FAQ: Find the points on the surface closest to the origin

1. What does it mean to find the points on the surface closest to the origin?

When we talk about finding the points on the surface closest to the origin, we are looking for the points on a given surface that are the shortest distance away from the origin point (0,0,0).

2. What is the significance of finding these points?

Finding the points on a surface closest to the origin is useful in various fields of science, such as physics, engineering, and computer graphics. It allows us to determine the minimum distance between a surface and a point, which can be helpful in optimizing designs or solving complex equations.

3. How do you calculate the points on a surface closest to the origin?

The calculation involves finding the normal vector of the surface at each point and then determining the point where the normal vector intersects the line connecting the origin and the point on the surface. This point will be the closest point on the surface to the origin.

4. Are there any real-world applications of finding the points on a surface closest to the origin?

Yes, there are many real-world applications for this concept. For example, in engineering, it can be used to find the minimum distance between a building and a fault line to ensure structural stability. In computer graphics, it can be used to render realistic 3D objects by calculating the distance between the object and the viewer.

5. Are there any limitations to this method of finding points?

One limitation is that it only applies to surfaces that can be defined by a mathematical equation. It may not be applicable to real-world objects that have irregular shapes or surfaces. Additionally, this method assumes that the origin is outside of the surface, so it may not work for surfaces that enclose the origin point.

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