This preview shows page 1. Sign up to view the full content.
Unformatted text preview: the distance between ( 1 , 0, 0) and the
2
2
plane 5x z y 0 is 1 5( 1 ) 1 0
2
s5 2 12 D 10 3
2 1
2 1 s3
6 3 s3 So the distance between the planes is s3 6.
EXAMPLE 10 In Example 3 we showed that the lines L1: x 1 t L2: x y 2s 2 y z 3t 3 4 z s t
3 4s are skew. Find the distance between them.
SOLUTION Since the two lines L 1 and L 2 are skew, they can be viewed as lying on two
parallel planes P1 and P2 . The distance between L 1 and L 2 is the same as the distance
between P1 and P2 , which can be computed as in Example 9. The common normal
vector to both planes must be orthogonal to both v1
1, 3, 1 (the direction of L 1 )
and v2
2, 1, 4 (the direction of L 2 ). So a normal vector is n v1 ij
13
21 v2 k
1
4 13 i 6j 5k If we put s 0 in the equations of L 2 , we get the point 0, 3,
tion for P2 is
13 x 0 6y 5z 3 3 0 or 3 on L 2 and so an equa 13x 6y 5z 3 0 If we now set t 0 in the equations for L 1 , we get the point 1, 2, 4 on P1 . So
the distance between L 1 and L 2 is the same as the distance from 1, 2, 4 to
13x 6y 5z 3 0. By Formula 9, this distance is
D  13.5 2
6 54 3 2 5 8
s230 2 0.53 Exercises 1. Determine whether each statement is true or false. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
( j)
(k) 13 1
6
2
s13 Two lines parallel to a third line are parallel.
Two lines perpendicular to a third line are parallel.
Two planes parallel to a third plane are parallel.
Two planes perpendicular to a third plane are parallel.
Two lines parallel to a plane are parallel.
Two lines perpendicular to a plane are parallel.
Two planes parallel to a line are parallel.
Two planes perpendicular to a line are parallel.
Two planes either intersect or are parallel.
Two lines either intersect or are parallel.
A plane and a line either intersect or are parallel. 2–5  Find a vector equation and parametric equations for
the line. 2. The line through the point 1, 0, vector 2 i 4j 3. The line through the point vector 3, 1, 3 and parallel to the 5k
2, 4, 10 and parallel to the 8 4. The line through the origin and...
View
Full
Document
This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.
 Fall '09
 hamrick

Click to edit the document details