Consider the set of eight vectors `V[ahati+bhatj+chatk:a,b,c in {1-1}]`. Three non-coplanar vectors cann be chosen from V in `2^(p)` ways, then p is

Options

A. 3
B. 5
C. 6
D. 4

Answer

B

Explanation

Eight vectors in `V` are
`vecu=hati+hatj+hatk, vecv=hati+hatj-hatk, vecw=hati-hatj+hatk, vec(alpha)=hati-hatj-hatk`
`vecu'=hatihatj-hatk,vecv'=-hati-hatj+hatk,vecw'=-hati-hatj+hatk,vec(alpha')=-hati+hatj+hatk`
Clearly `vecu,vecu',vecv,vecv',vecw,vecw'` and `vec(alpha),vec(alpha')` are four pairs of unlike parallel vectors. If we take any one pair out of these four pairs, and one vector form the remaining 6 vectors, we obtain three coplanar vectors.
`:.` Total number of ways of selecting three coplanar vectors
`=.^(6)C_(1)xx.^(4)C_(1)=24`
Also,
Number of waysof selecting 3 vectors out of given 8 vectors, `.^(8)C_(3)`
`:.` Number of ways of selecting 3 no -coplanar vectors
`.^(8)C_(3)-24=32=2^(5)`
Hence `n=5`