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Homework Statement
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.
Homework 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]
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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