hwsol_5 - Homework Assignment 5- (10.4)(10.5) - Solutions...

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Homework Assignment 5- (10.4)(10.5) - Solutions Page 824: 63, 64*, 65, 66*, 67, 68* 64. Area of the parallelogram with 2 adjacent sides formed by ! ! 2,1 " and ! 1, ! 3 " . ! ! "#! 1, ! 3 "$ i % j % k % ! 21 0 1 ! 30 $ 5 k % , the area $ 5 k % $ 5. 66. Area of the triangle with vertices ! 0,0,0 " , ! 0, ! " and ! 1, ! 3,0 " . u % $! 0, ! " and v % $! 1, ! " , u % # v % $ i % j % k % 0 ! 21 1 ! $ 3 i % & j % & 2 k % the area of the triangle $ 1 2 || u % # v % || $ 1 2 3 2 & 1 & 2 2 $ 1 2 14 $ 1. 87 68. Volume of the parallelepiped with 3 adjacent edges formed by ! 0, ! 1,0 " , ! 0,2, ! 1 " and ! 1,0,2 " . ! 0, ! ! 1 "$ i % j % k % 0 ! 10 02 ! 1 $ i % , the volume $ | ! 0, ! ! 1 " # ! " | $ | ! 1,0,0 " # ! " | $ 1 Page 834: 8. the line through P 0, 2, 1 , and Q 2, 0, 2 d % $ PQ $! 2, ! " parametric equations: x $ 2 t y ! 2 $ ! 2 t z ! 1 $ t ; symmetric equations: x 2 $ y ! 2 ! 2 $ z ! 1 10. P 2, 0, ! 1 , parallel to x $ ! 3 t , y $ 2 ! 4 t , z $ ! 6 d % $! ! 3, ! 4, 0 " parametric equations : x $ 2 & 3 t y $ ! 4 t z $ ! 1 ; symmetric equations: x ! 2 3 $ y ! 4 , z $ ! 1 12. P ! 1, 0, 0 , parallel to x & 1 ! 2 $ y 3 $ z ! 2 1
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d % $! ! 2, 3, 1 " parametric equations : x $ ! 1 ! 2 t y $ 3 t z $ t ; symmetric equations: x & 1 ! 2 $ y 3 $ z 14. P ! 3, 1, 0 , perpendicular to ! 0, ! 3,1 " and ! 4,2, ! 1 " d % $! 0, ! "#! ! 1 "$ i % j % k % 0 ! 31 42 ! 1 $ i % & 4 j % & 12 k % parametric equations: x $ ! 3 & t y $ 1 & 4 t z $ 12 t ; symmetric equations: x & 3 1 $ y ! 1 4 $ z 12 16. P 0, ! 2, 1 , normal to the plane y & 3 z $ 4 d % $! 0,1,3 " parametric equations: x $ 0 y $ ! 2 & t z $ 1 & 3 t ; symmetric equations: y & 2 1 $ z ! 1 3 , x $ 0 20. L 1 $ x $ 1 ! 2 t y $ 2 t z $ 5 ! t , L 2 $ x $ 3 & 2 s y $ ! 2 ! 2 s z $ 6 & s d % 1 $! ! 2,2, ! 1 " , d % 2 $! 2, ! 2,1 " , since d % 2 $ ! d % 1 , these two lines are parallel. 24. L 1 $ x $ 3 & t y $ 3 & 3 t z $ 4 ! t , L 2 $ x $ 2 ! s y $ 1 ! 2 s z $ 6 & 2 s d % 1 $! 1,3, ! 1 " , d % 2 $! ! 1, ! 2,2 " . Since d % 1 " cd % 2 , these two lines are not parallel. Check to see if two lines intersect: x :3 & t $ 2 ! s $ t $ ! 1 ! s y & 3 t $ 3 & 3 ! ! 1 ! s " $ 3 ! 3 ! 3 s $ ! 3 s $ 1 ! 2 s , s $ ! 1, t $ ! 1 ! ! ! 1 " $ 0, s $ ! 1 z :4 ! t $ 4 ! 0 $ 4, 6 & 2 ! ! 1 " $ 6 ! 2 $ 4 Two lines intersect at where s $ ! 1 and t $ 0orat 3, 3, 4 . 28. P ! ! 2,0,3 " , normal vector n % $! 4,3, ! 2 " equation of the plane: 4 ! x & 2 " & 3 y ! 2 ! z ! 3 " $ 0 2
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32. P ! 1, ! 2,1 " , Q ! 2, ! 1,0 " , R ! 3, ! 2,2 " PQ $! 1,1, ! 1 " , PR $! 2,0,1 " , n % $ PQ # PR $ i % j % k % 11 ! 1 201 $ i % ! 3 j % ! 2 k % equation of the plane: ! x ! 1 " ! 3 ! y & 2 " ! 2 ! z ! 1 " $ 0 36. P ! 3, ! " , parallel to x & 3 y ! 4 z $ 2 n % $! 1,3, ! 4 " , the equation of the plane: ! x ! 3 " & 3 ! y & 2 " ! 4 ! z ! 1 " $ 0 40. P ! 3,0, ! 1 " , perpendicular to the planes x & 2 y ! z $ 2 and 2 x ! z $ 1 n % 1 $!
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hwsol_5 - Homework Assignment 5- (10.4)(10.5) - Solutions...

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