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Ex 10.4, 8 - If either a = 0 or b = 0, then a x b = 0. Converse

Ex 10.4, 8 - Chapter 10 Class 12 Vector Algebra - Part 2

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Ex 10.4, 8 If either π‘Ž βƒ— = 0 βƒ— or 𝑏 βƒ— = 0 βƒ—, then π‘Ž βƒ— Γ— 𝑏 βƒ— = 0 βƒ— . Is the converse true? Justify your answer with an example. Converse : If π‘Ž βƒ— Γ— 𝑏 βƒ— = 0 βƒ—, then either π‘Ž βƒ— = 0 βƒ— or 𝑏 βƒ— = 0 βƒ— π‘Ž βƒ— Γ— 𝑏 βƒ— = |π‘Ž βƒ— ||𝑏 βƒ— | sin ΞΈ 𝑛 Μ‚ where, ΞΈ = angle between π‘Ž βƒ— and 𝑏 βƒ— 𝑛 Μ‚ = unit vector perpendicular to π‘Ž βƒ— πœ€ 𝑏 βƒ— Let π‘Ž βƒ— = 1𝑖 Μ‚ + 1𝑗 Μ‚ + 1π‘˜ Μ‚ & 𝑏 βƒ— = 2𝑖 Μ‚ + 2𝑗 Μ‚ + 2π‘˜ Μ‚ π‘Ž βƒ— Γ— 𝑏 βƒ— = |β– 8(𝑖 Μ‚&𝑗 Μ‚&π‘˜ Μ‚@1&1&1@2&2&2)| = 𝑖 Μ‚ (1 Γ— 2 βˆ’ 2 Γ— 1) βˆ’ 𝑗 Μ‚ (1 Γ— 2 βˆ’2 Γ— 1) + π‘˜ Μ‚ (1 Γ— 2 βˆ’ 2 Γ— 1) + π‘˜ Μ‚(1 Γ— 2 βˆ’ 2 Γ— 1) = 𝑖 Μ‚ (2 βˆ’ 2) βˆ’ 𝑗 Μ‚ (2 βˆ’2) + π‘˜ Μ‚ (2 βˆ’ 2) = 0𝑖 Μ‚ βˆ’ 0𝑗 Μ‚ + 0π‘˜ Μ‚ = 0 βƒ— Here, π‘Ž βƒ— β‰  0 βƒ— & 𝑏 βƒ—β‰  0 βƒ— But π‘Ž βƒ— Γ— 𝑏 βƒ— = 0 βƒ— Therefore, converse is not true.

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