Ex 10.5, 6 (Supplementary NCERT) - Chapter 10 Class 12 Vector Algebra

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Ex 10.5, 6 (Supplementary NCERT)Find x such that A (3, 2, 1), B = (4, x, 5), C(4, 2, 2) and D (6, 5, 1) are coplanar Four points A, B, C, D are coplanar if the three vectors ( ) , ( ) and ( ) are coplanar. i.e. [( ) , ( ) , ( ) ] = 0 A (3, 2, 1), B (4, x, 5) ( ) = (4 3) + (x 2) + (5 1) = 1 + (x 2) + 4 A (3, 2, 1), C(4, 2, 2) ( ) = (4 3) + (2 2) + ( 2 1) = 1 + 0 3 A (3, 2, 1), D (6, 5, 1) ( ) = (6 3) + (5 2) + ( 1 1) = 3 + 3 2 A (3, 2, 1), D (6, 5, 1) ( ) = (6 3) + (5 2) + ( 1 1) = 3 + 3 2 A (3, 2, 1), D (6, 5, 1) ( ) = (6 3) + (5 2) + ( 1 1) = 3 + 3 2 A (3, 2, 1), D (6, 5, 1) ( ) = (6 3) + (5 2) + ( 1 1) = 3 + 3 2 [( ) , ( ) , ( ) ] = | 8(1&( 2)&4@1&0& 3@3&3& 2)| 0 = 1[(0 2) (3 3) ] (x 2) [(1 2) (3 3)] + 4[(1 3) (3 0) ] 0 = 1[0 ( 9)] "(x 2)" [ 2 ( 9)]+4[3 0] 0 = 1(9) "(x 2)"( 2 + 9) + 4(3) 0 = 9 (x 2) (7) + 12 0 = 9 7x + 14 + 12 0 = 35 7x 7x = 35 x = 35/7 x = 5 Ex 10.5, 6 (Supplementary NCERT)Find x such that A (3, 2, 1), B = (4, x, 5), C(4, 2, 2) and D (6, 5, 1) are coplanar Four points A, B, C, D are coplanar if the three vectors ( ) , ( ) and ( ) are coplanar. i.e. [( ) , ( ) , ( ) ] = 0 A (3, 2, 1), B (4, x, 5) ( ) = (4 3) + (x 2) + (5 1) = 1 + (x 2) + 4 A (3, 2, 1), C(4, 2, 2) ( ) = (4 3) + (2 2) + ( 2 1) = 1 + 0 3 A (3, 2, 1), C(4, 2, 2) ( ) = (4 3) + (2 2) + ( 2 1) = 1 + 0 3 A (3, 2, 1), D (6, 5, 1) ( ) = (6 3) + (5 2) + ( 1 1) = 3 + 3 2 [( ) , ( ) , ( ) ] = | 8(1&( 2)&4@1&0& 3@3&3& 2)| 0 = 1[(0 2) (3 3) ] (x 2) [(1 2) (3 3)] + 4[(1 3) (3 0) ] 0 = 1[0 ( 9)] "(x 2)" [ 2 ( 9)]+4[3 0] 0 = 1(9) "(x 2)"( 2 + 9) + 4(3) 0 = 9 (x 2) (7) + 12 0 = 9 7x + 14 + 12 0 = 35 7x 7x = 35 x = 35/7 x = 5