Intersecting Line & Unit Sphere: Find Point of Intersection

[goonking]

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.

 

[vela]

You could try finding how close the line gets to the origin. If it's less than 1, it must intersect the unit sphere.

 

[goonking]

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?

 

[davidmoore63@y]

What's wrong with your approach? I used it and got the answer quickly.

 

[Ray Vickson]

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.

 

[goonking]

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.

 

[Mark44]

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.