Chapter 13

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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...
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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.

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