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Intersecting lines point of intersection.?
Determine whether the lines
L_1: x = 19 + 6 t, y = 20 + 6 t, z = 7 + t
and
L_2: x = -13 + 7 t, y = -14 + 8 t, z = -4 + 4 t
Point of intersection: (__, __, __)
1 Answer
- IsaacUSMLv 41 decade agoFavorite Answer
L1 and L2 intersect if and only if there is a point (a, b, c) on both lines. Consequently, L1 and L2 intersect if and only if there exist real numbers t1 and t2 such that
........19 + 6*t1 = -13 + 7*t2 and 20 + 6*t1 = -14 + 8*t2 and 7 + t1 = -4 + 4*t2
Solve for t1 and t2 using the first and second equations.
........19 + 6t1 = -13 + 7t2
........20 + 6t1 = -14 + 8t2
Subtract the first equation from the second to get 1 = -1 + t2. Therefore t2 = 2. Plug in 2 in place of t2 in the third equation to get 7 + t1 = - 4 + 4(2), so t1 = -3. We must now check that the values obtained for t1 and t2 do in fact satisfy all of the equations:
........19 + 6(-3) = 1 = -13 + 7(2)
........20 + 6(-3) = 2 = -14 + 8(2)
..........7 +..(-3) = 4 =.. -4 + 4(2)
Reading the middle column above, we see that L1 and L2 intersect at the point (1, 2, 4). If you'd like visual confirmation, please take a look at the supplied graphic on photobucket. It was constructed in Mathematica 6.0.
