Question
Sun March 15, 2015 By: Pragyashree Jain

find the equation of line passing through the point (-1,-1,2) and perpendicular to each of the planes: 2x+3y-3z=2 and
5x-4y+z=6.

Expert Reply
Vimala Ramamurthy
Mon March 16, 2015
L e t space a comma space b space a n d space c space b e space t h e space d i r e c t i o n space r a t i o s space o f space t h e space n o r m a l space t o space t h e space p l a n e. T h e space e q u a t i o n space o f space t h e space p l a n e space p a s sin g space t h r o u g h space open parentheses x subscript 1 comma y subscript 1 comma z subscript 1 close parentheses space i s a open parentheses x minus x subscript 1 close parentheses plus b open parentheses y minus y subscript 1 close parentheses plus c open parentheses z minus z subscript 1 close parentheses equals 0 T h e r e f o r e space t h e space e q u a t i o n space o f space t h e space p l a n e space p a s sin g space t h r o u g h space open parentheses negative 1 comma negative 1 comma 2 close parentheses space i s a open parentheses x plus 1 close parentheses plus b open parentheses y plus 1 close parentheses plus c open parentheses z minus 2 close parentheses equals 0... left parenthesis 1 right parenthesis  G i v e n space t h a t space t h e space r e q u i r e d space p l a n e space i s space p e r p e n d i c u l a r space t o space t h e space p l a n e s 2 x plus 3 y minus 3 z equals 2 space a n d space 5 x minus 4 y plus z equals 6 T h u s comma space w e space h a v e 2 a plus 3 b minus 3 c equals 0 a n d 5 a minus 4 b plus c equals 0 S o l v i n g space b y space c r o s s space m u l t i p l i c a t i o n comma space w e space h a v e comma fraction numerator a over denominator 3 cross times 1 minus open parentheses negative 4 close parentheses cross times open parentheses negative 3 close parentheses end fraction equals fraction numerator b over denominator 5 cross times open parentheses negative 3 close parentheses minus 2 cross times 1 end fraction equals fraction numerator c over denominator 2 cross times open parentheses negative 4 close parentheses minus 5 cross times 3 end fraction rightwards double arrow fraction numerator a over denominator 3 minus 12 end fraction equals fraction numerator b over denominator negative 15 minus 2 end fraction equals fraction numerator c over denominator negative 8 minus 15 end fraction rightwards double arrow fraction numerator a over denominator negative 9 end fraction equals fraction numerator b over denominator negative 17 end fraction equals fraction numerator c over denominator negative 23 end fraction equals k rightwards double arrow a equals 9 k semicolon b equals 17 k semicolon c equals 23 k s u b s t i t u t i n g space t h e space v a l u e s space o f space a comma space b space a n d space c space i n space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma 9 k open parentheses x plus 1 close parentheses plus 17 k open parentheses y plus 1 close parentheses plus 23 k open parentheses z minus 2 close parentheses equals 0 rightwards double arrow 9 open parentheses x plus 1 close parentheses plus 17 open parentheses y plus 1 close parentheses plus 23 open parentheses z minus 2 close parentheses equals 0 rightwards double arrow 9 x plus 9 plus 17 y plus 17 plus 23 z minus 46 equals 0 rightwards double arrow 9 x plus 17 y plus 23 z minus 20 equals 0 T h u s space t h e space e q u a t i o n space o f space t h e space r e q u i r e d space p l a n e space i s space 9 x plus 17 y plus 23 z minus 20 equals 0
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