Lines
in Space
Example: Determine whether the two
lines L1 and
L2 are parallel. L1 x
= 6 +
2t, y =
5 + 2t,
z = 7
+ 3t L2 x
= 7 +
3s, y =
5 + 3s,
z = 10 + 5s Note
the lines in “symmetric” form are: (x -
6)/2 = (y – 5)/2 =
(z – 7)/3 = t and (x – 7)/3 =
(y – 5)/ 3 = (z – 10)/5 =
s So, the vector parallel to
line L1 is
V1 = 2 i + 2 j
+
3 k and the vector parallel to
line L2 is
V2 = 3 i + 3 j +
5 k Since V1
is not a multiple of V2
, these vectors and the lines are
not parallel. (result) |
Next determine if the lines L1 and
L2 skew or
intersecting. If the lines intersect,
then they must have a common point at the point of intersection. If that is not the case,
then the lines are skew. So assume that the two lines intersect. Then: x
= 6 +
2t = 7
+ 3s eq (1) y
= 5 +
2t = 5
+ 3s eq (2) z
= 7 +
3t = 10
+ 5s eq (3) Subtract (2) from (3). The result is: 2
+ t =
5 + 2s or
t = 3
+ 2s So
2t = 6
+ 4s . Put this into (2) and solve for s. 5
+ 6 +
4s = 5
+ 3s, or
s = -6 Then put this result into t
= 3 + 2s
which yields t = -9 Now determine if these
values of s and t satisfy eq (1). 6
+ 2(-9) =?
7 + 3(-6)
which does not hold! So the lines L1 and
L2 do not
intersect. Therefore they are
skew. (result) |
Copyright © 2017 Richard C. Coddington
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