# Chapter 13

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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.

Ask a homework question - tutors are online