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In a four-dimensional space where unit v...

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

A

(a) `lamda=1`

B

(b) `mu=-(2)/(3)`

C

(c) `gamma=(2)/(3)`

D

(d) `delta=(1)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, D
`(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)`.
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