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Failure007
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Homework Statement
Find the vector, parametric and symmetric equations of a line that intersect both line 1 and line 2 at 90°.
L1 :
x = 4 + 2t
y = 8 + 3t
z = -1 - 4t
L2 :
x = 7 - 6t
y = 2+ t
z = -1 + 2t
Homework Equations
[tex]\vec{a}[/tex]1 - [tex]\vec{a}[/tex]2 + [tex]\vec{b}[/tex]1t - [tex]\vec{b}[/tex]2s
The Attempt at a Solution
=[4,8,-1] - [7,2,-1] + [2t, 3t, -4t] - [-6s, 1s, 2s]
=[-3, 6, 0] + [(2t + 6s), (3t-1s), (-4t-2s)]
=[(-3 + 2t +6s), (6 + 3t - s), (0 - 4t -2s)]
=[2, 3, -4] . [(-3 + 2t +6s), (6 + 3t - s), (0 - 4t -2s)]
=(-6 + 4t + 12s) + (18 + 9t -3s) + (0 +16t + 8s)
=12 + 29t +17s -------(equation 1)
=[-6, 1, 2] . [(-3 + 2t +6s), (6 + 3t - s), (0 - 4t -2s)]
=(18 -12t -36s) + (6 + 3t - s) + (0 -8t -4s)
=24 -17t -41s ----------(equation 2)
29t + 17s = -12
-17 -41s = -24
-493t -1189s = -696
-493t -289s =204
-900s = -900
Therefore, s = 1
29 + 17(1) = -12
29t = -29
Therefore, t=-1
At these values, the dot product of
[tex]\vec{b}[/tex]1 . ([tex]\vec{a}[/tex]1 - [tex]\vec{a}[/tex]2 + [tex]\vec{b}[/tex]1t - [tex]\vec{b}[/tex]2s)
and
[tex]\vec{b}[/tex]2 . ([tex]\vec{a}[/tex]1 - [tex]\vec{a}[/tex]2 + [tex]\vec{b}[/tex]1t - [tex]\vec{b}[/tex]2s)
both equal 0, meaning this should be perpendicular to both lines.
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I'm honestly not even sure if I'm on the right track with this but it seemed like it was working up until this point, however I'm not really sure how to derive the vector equation from this given information, so if anyone at all has some info that would point me in the right direction it would be greatly appreciated. Thanks to anyone and everyone willing to help.