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(716, #15) Determine whether the lines
and
are parallel, skew, or intersecting.
If they intersect, find the point of intersection.
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Solution:
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The parametric equations for
are:
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2) : x = 2 + s, y = -1 + 3s, z = 2s

The direction vectors for and are < 2, 4, 3 > and < 1, 3, 0 >. The lines are not parallel because their vectors are not proportional.

If the lines were to intersect we would have three equations in
two unknowns *s* and *t*:

4 + 2t = 2 + s

-5 + 4t = -1 + 3s

1 + 3t = 2s

## 2 = s - 2t -> -6 = -3s + 6t -> 2t = -10 -> t = -5 and s = -8 -4 = 3s - 4t -4 = 3s - 4t

## -4 = 3s - 4t -> -8 = 6s - 8t -> t = -11 and s = -16 1 = 2s - 3t 3 = -6s + 9t

Thus the lines do not intersect because different values of *t*
and *s* are obtained for solving the first and second, and
then the second and third equations.

Since the lines are not parallel and they do not intersect, then they must be skew.

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by: nl
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