In a four-dimensional space where unit vectors along the axes are `hati,hatj,hatk and hatl, and a_(1),a_(2),a_(3),a_(4) ` are four non-zero vectors such that no vector can be expressed as a linear combination of other `(lamda-1) (a_(1)-a_(2))+mu(a_(2)+a_(3))+gamma(a_(3)+a_(4)-2a_(2))+a_(3)+deltaa_(4)=0`, then

Options

A. (a) `lamda=1`
B. (b) `mu=-(2)/(3)`
C. (c) `gamma=(2)/(3)`
D. (d) `delta=(1)/(3)`

Answer

A, B, D

Explanation

`(lamda-1)(a_(1)-a_(2))+mu(a_(2)+a_(3))+gamma(a_(3)+a_(4)-2a_(2))+a_(3)+delta a_(4)=0`
`i.e., (lamda-1)a_(1)+(1-lamda+mu-2gamma)a_(2)+(mu+gamma+1)a_(3)+(gamma+delta)a_(4)=0`
Since, `a_(1),a_(2),a_(3) and a_(4)` are linearly independent, we have
`lamda-1=0,1-lamda+mu-2gamma=0`,
`mu+gamma+1=0 and gamma+delta=0`
i.e., `lamda=1,mu=2gamma,mu+gamma+1=0,gamma+delta=0`
Hence, `lamda=1,mu=-(2)/(3),gamma=-(1)/(3),delta=(1)/(3)`.