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Ex 10.3, 10 - If a, b, c are such, a + b is perpendicular to c

Ex 10.3, 10 - Chapter 10 Class 12 Vector Algebra - Part 2

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Ex 10.3, 10 If π‘Ž βƒ— = 2𝑖 Μ‚ + 2𝑗 Μ‚ + 3π‘˜ Μ‚, 𝑏 βƒ— = βˆ’π‘– Μ‚ + 2𝑗 Μ‚ + π‘˜ Μ‚ and 𝑐 βƒ— = 3𝑖 Μ‚ + 𝑗 Μ‚ are such that π‘Ž βƒ— +πœ†π‘ βƒ— is perpendicular to 𝑐 βƒ— , then find the value of πœ†.π‘Ž βƒ— = 2𝑖 Μ‚ + 2𝑗 Μ‚ + 3π‘˜ Μ‚ 𝑏 βƒ— = βˆ’π‘– Μ‚ + 2𝑗 Μ‚ + π‘˜ Μ‚ = βˆ’1𝑖 Μ‚ + 2𝑗 Μ‚ + 1π‘˜ Μ‚ 𝑐 βƒ— = 3𝑖 Μ‚ + 𝑗 Μ‚ = 3𝑖 Μ‚ + 1𝑗 Μ‚ + 0π‘˜ Μ‚ Now, (π‘Ž βƒ— + πœ†π‘ βƒ—) = (2𝑖 Μ‚ + 2𝑗 Μ‚ + 3π‘˜ Μ‚) + πœ† (-1𝑖 Μ‚ + 2𝑗 Μ‚ + 1π‘˜ Μ‚) = 2𝑖 Μ‚ + 2𝑗 Μ‚ + 3π‘˜ Μ‚ βˆ’ πœ†π‘– Μ‚ + 2πœ†π‘— Μ‚ + πœ†π‘˜ Μ‚ = (2 βˆ’ πœ†) 𝑖 Μ‚ + (2 + 2πœ†) 𝑗 Μ‚ + (3 + πœ†) π‘˜ Μ‚ Since (π‘Ž βƒ— + πœ†π‘ βƒ—) is perpendicular to 𝑐 βƒ— (π‘Ž βƒ— + πœ†π‘ βƒ—). 𝑐 βƒ— = 0 [(2βˆ’πœ†) 𝑖 Μ‚+(2+2πœ†) 𝑗 Μ‚+(3+πœ†)π‘˜ Μ‚ ] . (3𝑖 Μ‚ + 1𝑗 Μ‚ + 0π‘˜ Μ‚) = 0 (2 βˆ’ πœ†).3 + (2 + 2πœ†).1 + (3 + πœ† ).0 = 0 3.2 βˆ’ 3πœ† + 2 + 2πœ† + 0 = 0 6 – 3πœ† + 2 + 2πœ† = 0 8 βˆ’ πœ† = 0 πœ† = 8 ∴ πœ† = 8 (Dot product of perpendicular vectors is 0)

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