How to Find the Intersection Point of Two Lines in 3D Space

[Miike012]

Ive tried finding the intersection point that I labeled I.

first try:

Because I know the points A and M1 I found |AM1|
Then I said |AM1| = |AI| + |IM1| and proceeded to solve and came up with

-16 = 2(I₁)² +I₂)² + I₃)²) - 10I₁ - 8I₂ - 3I₃ +√(blah blah blah)

Second attempt:
Law of Cosines

First I found the angle between AB and MC
Second: Used law of cosines

|MI|² = ...

This way was going to be just as complicated so I stopped.

How can I solve this problem without using the fact that |M1| = 1/3|MC| and |IC| = 2/3|MC| or c.m?

 

[LCKurtz]

If you can't use the 2/3 thing, first find the mid points of a couple of sides then find the parametric equations of two medians (use different parameters, like $t$ and $s$). Then solve for where the two lines intersect.

 

[Miike012]

Are you saying find the equations of the two red lines?

Would I also get the same result if I found the equation of the two green lines?

or the green line that looks vertical and the red line not // to the green line that appears to be vertical?

 

[LCKurtz]

Are you saying find the equations of the two red lines?

Would I also get the same result if I found the equation of the two green lines?

or the green line that looks vertical and the red line not // to the green line that appears to be vertical?

The three medians of a triangle are concurrent. Use any two of the medians. You will always get the same answer for the intersection point. And there are only two medians drawn in your picture, each partially red and partially green.

 

[SammyS]

Maybe this graphic is more readable --- although it gets pixelated at high magnification.

 

[Miike012]

The equation I got for AM1 is

(I₁)² + (I₂)² + (I₃)² -2I₁ - 6I₃ = -16

Is this what the equation should look like?

 

[SammyS]

The equation I got for AM1 is

(I₁)² + (I₂)² + (I₃)² -2I₁ - 6I₃ = -16

Is this what the equation should look like?

If you can't use the 2/3 thing, first find the mid points of a couple of sides then find the parametric equations of two medians (use different parameters, like $t$ and $s$). Then solve for where the two lines intersect.

Your equation doesn't have a parameter, such as s or t or whatever.

Therefore, I will presume that your equation is not what LCKurtz suggested.

*****

Actually, if you find the coordinates of the point, M, you can find the coordinates of I, by using the 2/3's thing.

That's not what LCKurtz suggested.

He suggested finding a parametric equation for the line determined by points A and M1 & finding a parametric equation for the line determined by points C and M, then finding the point at which they intersect.

 

[Miike012]

I₂ = √[ -(I₁)² -(I₃)² + 2I₁ + 6I₃ - 16]

Better?

 

[LCKurtz]

If you can't use the 2/3 thing, first find the mid points of a couple of sides then find the parametric equations of two medians (use different parameters, like $t$ and $s$). Then solve for where the two lines intersect.

I₂ = √[ -(I₁)² -(I₃)² + 2I₁ + 6I₃ - 16]

Better?

It doesn't look better to me; I can't make heads or tails of whatever that is supposed to be.

If you ask for help, why don't you try some of the suggestions? Above I suggested you find the mid points of two sides. Have you done that? You certainly haven't shown us.

Then I suggested you write the parametric equations of two of the medians. Have you done that? Do you know how? Or am I just wasting my time trying to help you?

 

[Miike012]

|AM|² = (|AI| + |IM|)²

Where I is the point of intersection
Midpoint M is (1,2,3)

(|AI| + |IM|)² = |AI|² +2|AI||IM| + |IM|²

|AI||IM| = (AI dot IM)/cosθ = (AI dot IM)

(|AI| + |IM|)² = |AI|² +2(AI dot IM) + |IM|²

This is how I proceeded to solve for AM

Is this wrong??

 

[LCKurtz]

|AM|² = (|AI| + |IM|)²

Where I is the point of intersection
Midpoint M is (1,2,3)

(|AI| + |IM|)² = |AI|² +2|AI||IM| + |IM|²

|AI||IM| = (AI dot IM)/cosθ = (AI dot IM)

(|AI| + |IM|)² = |AI|² +2(AI dot IM) + |IM|²

This is how I proceeded to solve for AM

Is this wrong??

Who are you responding to, me? Quote the message to which you are responding. Midpoint M is not (1,2,3). And your whole idea is so on the wrong track it isn't worth explaining.

If you want to continue this discussion with me, respond to post #9 by quoting it and by answering the questions I asked. Otherwise I am done wasting my time with this thread.

 

[Miike012]

It doesn't look better to me; I can't make heads or tails of whatever that is supposed to be.

If you ask for help, why don't you try some of the suggestions? Above I suggested you find the mid points of two sides. Have you done that? You certainly haven't shown us.

Midpoint between points A and B is:
M(5/2,5/2,0)
and
M1(1,2,3)

Then I suggested you write the parametric equations of two of the medians. Have you done that? Do you know how? Or am I just wasting my time trying to help you?

I do not know how to find the parametric equations of the medians[/QUOTE]

 

[LCKurtz]

It doesn't look better to me; I can't make heads or tails of whatever that is supposed to be.

If you ask for help, why don't you try some of the suggestions? Above I suggested you find the mid points of two sides. Have you done that? You certainly haven't shown us.

Then I suggested you write the parametric equations of two of the medians. Have you done that? Do you know how? Or am I just wasting my time trying to help you?

Midpoint between points A and B is: M(5/2,5/2,0) and M1(1,2,3). I do not know how to find the parametric equations of the medians.

Check the third component of M1.

Do you know how to write the equation of a straight line between two points in 3D? Isn't that in your text? What course are you taking where this problem arose?

 

[Miike012]

Check the third component of M1.
Third component is 3/2

Do you know how to write the equation of a straight line between two points in 3D? Isn't that in your text? What course are you taking where this problem arose?

I haven't learned how to write an equation in 3D. I was supposed to find the intersection point using center of mass but I wanted to try to find it another way. I am in calc 3 and have not yet learned how to write an eqatuion in 3D

 

[LCKurtz]

I haven't learned how to write an equation in 3D. I was supposed to find the intersection point using center of mass but I wanted to try to find it another way. I am in calc 3 and have not yet learned how to write an eqatuion in 3D

Calc III usually starts with vectors, planes, and lines, not necessarily in that order. Apparently you are at the beginning of the course and not quite there yet. Why don't you wait a few days until you have covered lines in 3D. Then you will likely find it easy to follow my suggestions.