Points 5 3 2 direction vector v direction numbers

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Unformatted text preview: L 3 are parallel. 19. At the point of intersection, the coordinates for one line equal the corresponding coordinates for the other line. Thus, (i) 4t 2 2s 2, (ii) 3 2s 3, and (iii) t 1 s 1. From (ii), we find that s 0 and consequently, from (iii), t 0. Letting s t 0, we see that equation (i) is satisfied and therefore the two lines intersect. Substituting zero for s or for t, we obtain the point (2, 3, 1 . u v cos 4i 2i k 2j k 81 17 9 (First line) (Second line) 7 3 17 7 17 51 uv uv 21. Writing the equations of the lines in parametric form we have x x 3t 1 4s y y 2 2 t s z z 1 3 t 3s. 17 7 2 s. Solving this system yields t For the coordinates to be equal, 3t 1 4s and 2 t When using these values for s and t, the z coordinates are not equal. The lines do not intersect. 23. x y z 2t 5t t 3 2 1 x y z s 2s 2s 8 1 1 x −8 10 8 6 4 2 −2 4 6 8 and s 11 7. 7 4 z Point of intersection: 7, 8, 10 y (7, 8, − 1) 246 25. 4x Chapter 10 3y \ Vectors and the Geometry of Space 6 \ \ 6z 0, 0, 0, (a) P PQ 1 ,Q \ 0, 2, 0 , R 3, 4, 0 3, 4, 1 (b) PQ PR 2, 1 , PR i 0 3 j 2 4 k 1 0 4, 3, 6 The components of the cross product are proportional to the coefficients of the variables in the equation. The cross product is parallel to the normal vector. 27. Point: 2, 1, 2 n 1x x 2 i 2 0 1, 0, 0 0y 1 0z 2 0 29. Point: 3, 2, 2 Normal vector: n 2x 2x 3 3y 3y z 10 2 2i 3j 1z k 2 0 31. Point: 0, 0, 6 Normal vector: n 1x x x y y 2z 0 2z 1y 12 12 0 0 i j 2z 2k 6 0 33. Let u be the vector from 0, 0, 0 to 1, 2, 3 : u i 2j 3k Let v be the vector from 0, 0, 0 to v 2i 3j 3k Normal vector: u v i 1 2 3i 3x 3x 9y 0 7z 9y 0 0 7z 0 j 2 3 2, 3, 3 : k 3 3 9j 0 7k 35. Let u be the vector from 1, 2, 3 to 3, 2, 1 : u Let v be the vector from 1, 2, 3 to Normal vector: 4x 4x 1 3y 3y 4z 1 2u 2i 2k 2i 3j 4k 4j k 1, j 0 4 0 2, 2 : v k 1 1 4i v 4z 3 i 1 2 2 10 37. 1, 2, 3 , Normal vector: v k, 1 z 3 0, z 3 39. The direction vectors for the lines are u v 3i 4j k. Normal vector: u v i 2 3 j 1 4 k 1 1 1, 5, 1 0 2i j k, 5i j k Point of intersection of...
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This note was uploaded on 05/18/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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