Ex 10.3, 10

Transcript

Ex 10.3, 10 If 𝑎﷯ = 2 𝑖﷯ + 2 𝑗﷯ + 3 𝑘﷯, 𝑏﷯ = − 𝑖﷯ + 2 𝑗﷯ + 𝑘﷯ and 𝑐﷯ = 3 𝑖﷯ + 𝑗﷯ are such that 𝑎﷯ +𝜆 𝑏﷯ is perpendicular to 𝑐﷯ , then find the value of 𝜆. 𝑎﷯ = 2 𝑖﷯ + 2 𝑗﷯ + 3 𝑘﷯ 𝑏﷯ = − 𝑖﷯ + 2 𝑗﷯ + 𝑘﷯ = −1 𝑖﷯ + 2 𝑗﷯ + 1 𝑘﷯ 𝑐﷯ = 3 𝑖﷯ + 𝑗﷯ = 3 𝑖﷯ + 1 𝑗﷯ + 0 𝑘﷯ ( 𝑎﷯ + 𝜆 𝑏﷯) = (2 𝑖﷯ + 2 𝑗﷯ + 3 𝑘﷯) + 𝜆 (-1 𝑖﷯ + 2 𝑗﷯ + 1 𝑘﷯) = 2 𝑖﷯ + 2 𝑗﷯ + 3 𝑘﷯ − 𝜆 𝑖﷯ + 2𝜆 𝑗﷯ + 𝜆 𝑘﷯ = (2 − 𝜆) 𝑖﷯ + (2 + 2𝜆) 𝑗﷯ + (3 + 𝜆) 𝑘﷯ If ( 𝑎﷯ + 𝜆 𝑏﷯) is perpendicular to 𝑐﷯ ( 𝑎﷯ + 𝜆 𝑏﷯). 𝑐﷯ = 0 2−𝜆﷯ 𝑖﷯+ 2+2𝜆﷯ 𝑗﷯+(3+𝜆) 𝑘﷯﷯ . (3 𝑖﷯ + 1 𝑗﷯ + 0 𝑘﷯) = 0 (2 - 𝜆).3 + (2 + 2𝜆).1 + (3 + 𝜆 ).0 = 0 3.2 − 3𝜆 + 2 + 2𝜆 + 0 = 0 6 – 3𝜆 + 2 + 2𝜆 = 0 8 − 𝜆 = 0 𝜆 = 8 ∴ 𝜆 = 8