Examples

Chapter 10 Class 12 Vector Algebra
Serial order wise

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class

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Example 29 (Supplementary NCERT) Show that the four points A, B, C and D with position vectors 4π Μ + 5π Μ + π Μ, β(π Μ + π Μ), 3π Μ + 9π Μ + 4π Μ & β4π Μ + 4π Μ + 4π Μ, respectively coplanar Four points A, B, C, D are coplanar if the three vectors (π΄π΅) β , (π΄πΆ) β and (π΄π·) β are coplanar. i.e. [(π¨π©) β, (π¨πͺ) β, (π¨π«) β ] = 0 A (4π Μ + 5π Μ + π Μ) B (βπ Μ β π Μ) (π¨π©) β = (0π Μ β π Μ β π Μ) β (4π Μ + 5π Μ + π Μ) = β4π Μ β 6π Μ β 2π Μ A (4π Μ + 5π Μ + π Μ) C (3π Μ + 9π Μ + 4π Μ) (π¨πͺ) β = (3π Μ + 9π Μ + 4π Μ) β (4π Μ + 5π Μ + π Μ) = βπ Μ + 4π Μ + 3π Μ A (4π Μ + 5π Μ + π Μ) D (β4π Μ + 4π Μ + 4π Μ) (π¨π«) β = (β4π Μ + 4π Μ + 4π Μ) β (4π Μ + 5π Μ + π Μ) = β8π Μ β π Μ + 3π Μ [(π΄π΅) β, (π΄πΆ) β, (π΄π·) β ] = |β 8(β4&β6&β2@β1&4&3@β8&β1&3)| = β4[(4Γ3)β(β1Γ3) ] β (β6) [(β1Γ3)β(β8Γ3)] + (β2)[(β1Γβ1)β(β8Γ4) ] = β4 [12+3]+6[β3+24]β2[1+32] = β4 (15) + 6 (21) β 2 (33) = β60 + 126 β 66 = β126+ 126 = 0 β΄[(π΄π΅) β, (π΄πΆ) β, (π΄π·) β ] = 0 Therefore, points A, B, C and D are coplanar.