Last updated at May 29, 2018 by Teachoo

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Misc 2 (Method 1) If 1 , 1 , 1 and 2 , 2 , 2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are 1 2 2 1 , 1 2 2 1 , 1 2 2 1 . We know that is perpendicular to both & So, required line is cross product of lines having direction cosines 1 , 1 , 1 and 2 , 2 , 2 Required line = 1 1 1 2 2 2 = ( 1 2 2 1 ) ( 1 2 2 1 ) + ( 1 2 2 1 ) = ( 1 2 2 1 ) ( 2 1 1 2 ) + ( 1 2 2 1 ) Hence, direction cosines = 1 2 2 1 , 1 2 2 1 , 1 2 2 1 Direction cosines of the line perpendicular to both of these are 1 2 2 1 , 1 2 2 1 , 1 2 2 1 . Hence proved Misc 2 (Method 2) If 1 , 1 , 1 and 2 , 2 , 2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are 1 2 2 1 , 1 2 2 1 , 1 2 2 1 . Two lines having direction cosines 1 , 1 , 1 and 2 , 2 , 2 are mutually perpendicular if + + = Let a line having direction cosines 3 , 3 , 3 be perpendicular to both the given lines. So, + + = & + + = From (2) and (3), So, l3 = (m1n2 m2 n1) m3 = (n1l2 n2 l1) n3 = (l1m2 l2 m1) We have to find the value of Also, we know that the sum of square of direction cosines of a line is 1. 1 2 + 1 2 + 1 2 = 1 2 2 + 2 2 + 2 2 = 1 & 3 2 + 3 2 + 3 2 = 1 Substituting values of l3 , m3 and n3 in (6) 2(m1n2 m2 n1)2 + 2 (n1l2 n2 l1)2 + 2 (l1m2 l2 m1)2 = 1 2 (m1n2 m2 n1)2 + (n1l2 n2 l1)2 + (l1m2 l2 m1)2 = 1 2 = + + Solving denominator, 1 2 2 1 2 + 2 1 1 2 2 + 1 2 2 1 2 = 1 2 2 2 + 2 2 1 2 + 2 2 1 2 + 1 2 2 2 + 1 2 2 2 + 2 2 1 2 2 1 2 2 1 2 2 1 1 2 2 2 2 2 1 Method 1: Now, using equations (1), (4) and (5), i.e. 1 2 + 1 2 + 1 2 =0 (1) 2 2 + 2 2 + 2 2 = 1 (2) & 3 2 + 3 2 + 3 2 = 1 (3) Solving (l12 + m12 + n12) (l22 + m22 + n22) (l1l2 + m1m2 + n1n2)2 = (1)(1) (0)2 = 1 0 = 1 Solving again (l12 + m12 + n12) (l22 + m22 + n22) (l1l2 + m1m2 + n1n2)2 = l12 (l22 + m22 + n22) + m12 (l22 + m22 + n22) + n12 (l22 + m22 + n22) (l12l22 + m12m22 + n12n22 + 2l1l2m1m2 + 2m1m2n1n2 + 2l1l2n1n2) = l12l22 + l12m22 + l12n22 + m12l22 + m12m22 +m12n22 + n12l22 + n12m22 + n12 n22 l12l22 m12m22 n12n22 2l1l2m1m2 2 m1m2n1n2 2 l1l2n1n2 = l12m22 + l12n22 + m12l22 +m12n22 + n12l22 + n12m22 2l1l2m1m2 2 m1m2n1n2 2 l1l2n1n2 Equation (9) is same as (7) And from equation (8) 1 2 2 1 2 + 2 1 1 2 2 + 1 2 2 1 2 = 1 1 1 2 2 1 2 + 2 1 1 2 2 + 1 2 2 1 2 = 1 2 = 1 = 1 Therefore, the direction cosines are So, l3 = (m1n2 m2 n1) = 1 (m1n2 m2 n1) = m1n2 m2 n1 m3 = (n1l2 n2 l1) = 1 (n1l2 n2 l1) = n1l2 n2 l1 n3 = (l1m2 l2 m1) = 1 (l1m2 l2 m1) = l1m2 l2 m1 Hence proved Method 2: = 1 2 2 2 + 2 2 1 2 + 2 2 1 2 + 1 2 2 2 + 1 2 2 2 + 2 2 1 2 2 1 2 2 1 2 2 1 1 2 2 2 2 2 1 Adding and subtracting 1 2 2 2 , 1 2 2 2 and 1 2 2 2 , = 1 2 ( 2 2 + 2 2 + 2 2 ) 1 2 2 2 + 1 2 ( 2 2 + 2 2 + 2 2 ) 1 2 2 2 + 1 2 ( 2 2 + 2 2 + 2 2 ) 1 2 2 2 2 1 2 2 1 2 2 1 1 2 2 2 2 2 1 = 1 2 (1) 1 2 2 2 + 1 2 (1) 1 2 2 2 + 1 2 (1) 1 2 2 2 2 1 2 2 1 2 2 1 1 2 2 2 2 2 1 = ( 1 2 + 1 2 + 1 2 ) 1 2 2 2 + 1 2 2 2 + 1 2 2 2 +2 1 2 2 1 +2 2 1 1 2 + 2 2 2 2 1 = 1 1 2 2 2 + 1 2 2 2 + 1 2 2 2 +2 1 2 2 1 +2 2 1 1 2 + 2 2 2 2 1 = 1 (l1l2 + m1m2 + n1n2)2 = 1 02 = 1 0 = 1 1 2 2 1 2 + 2 1 1 2 2 + 1 2 2 1 2 = 1 1 1 2 2 1 2 + 2 1 1 2 2 + 1 2 2 1 2 = 1 2 = 1 = 1 Therefore, the direction cosines are So, l3 = (m1n2 m2 n1) = 1 (m1n2 m2 n1) = m1n2 m2 n1 m3 = (n1l2 n2 l1) = 1 (n1l2 n2 l1) = n1l2 n2 l1 n3 = (l1m2 l2 m1) = 1 (l1m2 l2 m1) = l1m2 l2 m1 Hence proved

Miscellaneous

Misc 1
Important

Misc 2 You are here

Misc 3

Misc 4 Important

Misc 5 Important

Misc 6 Important

Misc 7

Misc 8 Important

Misc 9 Important

Misc 10

Misc 11 Important

Misc 12 Important

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important

Misc 17 Important

Misc 18 Important

Misc 19 Important

Misc 20 Important

Misc 21 Important

Misc 22 Important

Misc 23 Important

Chapter 11 Class 12 Three Dimensional Geometry

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.