Intersecting Line & Unit Sphere: Find Point of Intersection

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Homework Help Overview

The discussion revolves around determining whether a line defined by two points intersects with a unit sphere. The original poster presents their approach using parametric equations and the equation of the sphere, expressing concerns about the complexity of the calculations involved.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss using the parametric form of the line and substituting into the sphere's equation. There are suggestions to consider the distance from the line to the origin to assess potential intersection. Some participants question the validity of the original poster's approach and offer alternative perspectives.

Discussion Status

The conversation is ongoing, with participants providing feedback on the original poster's method and exploring different ways to analyze the problem. There is no clear consensus on the best approach yet, but various lines of reasoning are being examined.

Contextual Notes

There are mentions of potential errors in the original poster's mathematical setup, specifically regarding the equation of the sphere. Additionally, the ability to edit posts is noted as a constraint affecting the discussion.

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Homework Statement


Does the line through the points (−1, −1, −2) and (1, 2, 1) intersect the unit sphere? If so, find the point(s) of intersection.

Homework Equations

The Attempt at a Solution


do i also use r = r0 + vt but instead , use equation of sphere this time?

so it would be:

v=<2,3,3>

then using (-1,-1,-2) as the point

x = -1 + 2t y = -1 + 3t z=-2 + 3t

plugging these into x2 + y2 + z2 = 1

then solve for t

but then the math gets way too long and just doesn't seem to be the correct approach.
 
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You could try finding how close the line gets to the origin. If it's less than 1, it must intersect the unit sphere.
 
vela said:
You could try finding how close the line gets to the origin. If it's less than 1, it must intersect the unit sphere.
do I need to use scalar projection for this?
 
What's wrong with your approach? I used it and got the answer quickly.
 
goonking said:

Homework Statement


Does the line through the points (−1, −1, −2) and (1, 2, 1) intersect the unit sphere? If so, find the point(s) of intersection.

Homework Equations

The Attempt at a Solution


do i also use r = r0 + vt but instead , use equation of sphere this time?

so it would be:

v=<2,3,3>

then using (-1,-1,-2) as the point

x = -1 + 2t y = -1 + 3t z=-2 + 3t

plugging these into x2 + y2 + z2 = 1

then solve for t

but then the math gets way too long and just doesn't seem to be the correct approach.

The math does not get way too long; it is a bit messy, but sometimes that is how things are.
 
Ray Vickson said:
The math does not get way too long; it is a bit messy, but sometimes that is how things are.
whoops, there seems to be an error from the copy paste, it should be x2 + y2 + z2 = 1 and not x2 + y2 + z2 = 1. I will fix it right now.

hmmm, I cannot seem to edit my OP.
 
goonking said:
whoops, there seems to be an error from the copy paste, it should be x2 + y2 + z2 = 1 and not x2 + y2 + z2 = 1. I will fix it right now.

hmmm, I cannot seem to edit my OP.
You can edit your post provided you do so within a short period of time. I have fixed your post for you.
 

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