(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Two lines in space are in the same plane. Line AB passes through points [tex]A(x,y,z)[/tex] and [tex]B(x,y,z)[/tex], and line CD passes through points [tex]C(x,y,z)[/tex] and [tex]D(x,y,z)[/tex]. Determine if these two lines are parallel. If they are not, determine the x,y,z coordinates where these two lines intersect.

2. Relevant equations

The parametric equations for lines passing through points [tex]A[/tex] and [tex]B[/tex] are:

[tex]x = A_{x} + (B_{x}-A_{x})t[/tex]

[tex]y = A_{y} + (B_{y}-A_{y})t[/tex]

[tex]x = A_{z} + (B_{z}-A_{z})t[/tex]

Symmetric equations for the same point:

[tex]\frac{x - A_{x}}{(B_{x}-A_{x})} = \frac{y - A_{y}}{(B_{y}-A_{y})} = \frac{z - A_{z}}{(B_{z}-A_{z})}[/tex]

3. The attempt at a solution

I tried setting

[tex]x = A_{x} + (B_{x}-A_{x})t[/tex]

and

[tex]x = C_{x} + (D_{x}-C_{x})t[/tex]

equal. This resulted in:

[tex]t = \frac{C_{x}-A_{x}}{(B_{x}-A_{x})-(D_{x}-C_{x})}[/tex]

When I worked this out, I got numbers that did not seem to fit. The symmetric equations did not seem to fit either. I also tried writing the equation down as:

[tex]z(x,y) = Ex + Fy + G[/tex]

but this is for a surface.

I have a feeling the answer is pretty simple, but for some reason I'm not finding it it. Any help is appreciated.

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# Homework Help: Intersection of Two Coplanar Lines in 3 Dimensions

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