Vector product - Area

Chapter 10 Class 12 Vector Algebra
Concept wise

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Ex 10.4, 9 Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5). A (1, 1, 2) , B (2, 3, 5) , C (1, 5, 5) Area of triangle ABC is 1/2 |(π΄π΅) β Γ (π΄πΆ) β | A (1, 1, 2) B (2, 3, 5) (π΄π΅) β = (2 β 1) π Μ + (3 β 1) π Μ + (5 β 2) π Μ = 1π Μ + 2π Μ + 3π Μ A (1, 1, 2) C (1, 5, 5) (π΄πΆ) β = (1 β 1) π Μ + (5 β 1) π Μ + (5 β 2) π Μ = 0π Μ + 4π Μ + 3π Μ (π¨π©) β Γ (π¨πͺ) β = |β 8(π Μ&π Μ&π Μ@1&2&3@0&4&3)| = π Μ (2Γ3β4Γ3 )βπ Μ (1Γ3β0Γ3 )+π Μ (1Γ4β0Γ2 ) = π Μ (6β12)βπ Μ(3β0) + π Μ (4β0) = β6π Μ β 3π Μ + 4π Μ Magnitude of (π΄π΅) β Γ (π΄πΆ) β = β((β6)2+(β3)2+42) |(π΄π΅) β" Γ " (π΄πΆ) β | = β(36+9+16) = β61 Area of triangle ABC = 1/2 |(π΄π΅) β" Γ " (π΄πΆ) β | = 1/2 Γ β61 = βππ/π