If the vectors `alpha hat(i) + alpha hat(j) + lambda hat(k), hat(i) + hat(k) and lambda hat(i) + lambda hat(j) + beta hat(k) ` lie on a plane, where `alpha, beta and lambda` are distinct non-negative numbers, then `lambda ` is

Options

A. Arithmetic mean of `alpha and beta`
B. Geometric mean of `alpha and beta`
C. Harmonic mean of `alpha and beta`
D. None of the above

Answer

B

Explanation

If three vectors are co-planar.
`rArr |(alpha, alpha, gamma), (1,0,1),(gamma ,gamma, beta)|=0`
`rArr alpha[0-gamma]-alpha[beta-gamma]+gamma[gamma-0]=0`
`rArr -alpha gamma-alpha beta+alpha gamma+gamma^(2)=0`
`rArr gamma^(2)=alpha beta`
`rArr ` So `alpha,beta,gamma` are in G.P.